Astral Dataset
Collection
Synthetic Math dataset and Benchmark • 4 items • Updated • 1
id int64 1 981 | problem stringlengths 97 5.29k | raw_html stringlengths 121 5.63k | url stringlengths 34 36 | answer stringlengths 1 29 |
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1 | If we list all the natural numbers below $10$ that are multiples of $3$ or $5$, we get $3, 5, 6$ and $9$. The sum of these multiples is $23$.
Find the sum of all the multiples of $3$ or $5$ below $1000$. | <p>If we list all the natural numbers below $10$ that are multiples of $3$ or $5$, we get $3, 5, 6$ and $9$. The sum of these multiples is $23$.</p>
<p>Find the sum of all the multiples of $3$ or $5$ below $1000$.</p> | https://projecteuler.net/problem=1 | 233168 |
2 | Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with $1$ and $2$, the first $10$ terms will be:
$$1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots$$
By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued ter... | <p>Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with $1$ and $2$, the first $10$ terms will be:
$$1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots$$</p>
<p>By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-v... | https://projecteuler.net/problem=2 | 4613732 |
3 | The prime factors of $13195$ are $5, 7, 13$ and $29$.
What is the largest prime factor of the number $600851475143$? | <p>The prime factors of $13195$ are $5, 7, 13$ and $29$.</p>
<p>What is the largest prime factor of the number $600851475143$?</p> | https://projecteuler.net/problem=3 | 6857 |
4 | A palindromic number reads the same both ways. The largest palindrome made from the product of two $2$-digit numbers is $9009 = 91 \times 99$.
Find the largest palindrome made from the product of two $3$-digit numbers. | <p>A palindromic number reads the same both ways. The largest palindrome made from the product of two $2$-digit numbers is $9009 = 91 \times 99$.</p>
<p>Find the largest palindrome made from the product of two $3$-digit numbers.</p> | https://projecteuler.net/problem=4 | 906609 |
5 | $2520$ is the smallest number that can be divided by each of the numbers from $1$ to $10$ without any remainder.
What is the smallest positive number that is evenly divisibledivisible with no remainder by all of the numbers from $1$ to $20$? | <p>$2520$ is the smallest number that can be divided by each of the numbers from $1$ to $10$ without any remainder.</p>
<p>What is the smallest positive number that is <strong class="tooltip">evenly divisible<span class="tooltiptext">divisible with no remainder</span></strong> by all of the numbers from $1$ to $20$?</p... | https://projecteuler.net/problem=5 | 232792560 |
6 | The sum of the squares of the first ten natural numbers is,
$$1^2 + 2^2 + ... + 10^2 = 385.$$
The square of the sum of the first ten natural numbers is,
$$(1 + 2 + ... + 10)^2 = 55^2 = 3025.$$
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is $3025 - 385... | <p>The sum of the squares of the first ten natural numbers is,</p>
$$1^2 + 2^2 + ... + 10^2 = 385.$$
<p>The square of the sum of the first ten natural numbers is,</p>
$$(1 + 2 + ... + 10)^2 = 55^2 = 3025.$$
<p>Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum... | https://projecteuler.net/problem=6 | 25164150 |
7 | By listing the first six prime numbers: $2, 3, 5, 7, 11$, and $13$, we can see that the $6$th prime is $13$.
What is the $10\,001$st prime number? | <p>By listing the first six prime numbers: $2, 3, 5, 7, 11$, and $13$, we can see that the $6$th prime is $13$.</p>
<p>What is the $10\,001$st prime number?</p> | https://projecteuler.net/problem=7 | 104743 |
8 | The four adjacent digits in the $1000$-digit number that have the greatest product are $9 \times 9 \times 8 \times 9 = 5832$.
73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
1254069874715852386305071569329096329... | <p>The four adjacent digits in the $1000$-digit number that have the greatest product are $9 \times 9 \times 8 \times 9 = 5832$.</p>
<p class="monospace center">
73167176531330624919225119674426574742355349194934<br>
96983520312774506326239578318016984801869478851843<br>
858615607891129494954595017379583319528532088055... | https://projecteuler.net/problem=8 | 23514624000 |
9 | A Pythagorean triplet is a set of three natural numbers, $a \lt b \lt c$, for which,
$$a^2 + b^2 = c^2.$$
For example, $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.
There exists exactly one Pythagorean triplet for which $a + b + c = 1000$.
Find the product $abc$. | <p>A Pythagorean triplet is a set of three natural numbers, $a \lt b \lt c$, for which,
$$a^2 + b^2 = c^2.$$</p>
<p>For example, $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.</p>
<p>There exists exactly one Pythagorean triplet for which $a + b + c = 1000$.<br>Find the product $abc$.</p> | https://projecteuler.net/problem=9 | 31875000 |
10 | The sum of the primes below $10$ is $2 + 3 + 5 + 7 = 17$.
Find the sum of all the primes below two million. | <p>The sum of the primes below $10$ is $2 + 3 + 5 + 7 = 17$.</p>
<p>Find the sum of all the primes below two million.</p> | https://projecteuler.net/problem=10 | 142913828922 |
11 | In the $20 \times 20$ grid below, four numbers along a diagonal line have been marked in red.
08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
52 70 95 23 04 60 11 42 69 24 68 56 01 32 ... | <p>In the $20 \times 20$ grid below, four numbers along a diagonal line have been marked in red.</p>
<p class="monospace center">
08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08<br>
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00<br>
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65<br... | https://projecteuler.net/problem=11 | 70600674 |
12 | The sequence of triangle numbers is generated by adding the natural numbers. So the $7$th triangle number would be $1 + 2 + 3 + 4 + 5 + 6 + 7 = 28$. The first ten terms would be:
$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$$
Let us list the factors of the first seven triangle numbers:
$$\begin{align}
\mathbf 1 &\col... | <p>The sequence of triangle numbers is generated by adding the natural numbers. So the $7$<sup>th</sup> triangle number would be $1 + 2 + 3 + 4 + 5 + 6 + 7 = 28$. The first ten terms would be:
$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$$</p>
<p>Let us list the factors of the first seven triangle numbers:</p>
$$\begin... | https://projecteuler.net/problem=12 | 76576500 |
13 | Work out the first ten digits of the sum of the following one-hundred $50$-digit numbers.
37107287533902102798797998220837590246510135740250
46376937677490009712648124896970078050417018260538
74324986199524741059474233309513058123726617309629
91942213363574161572522430563301811072406154908250
230675882075393461711... | <p>Work out the first ten digits of the sum of the following one-hundred $50$-digit numbers.</p>
<div class="monospace center">
37107287533902102798797998220837590246510135740250<br>
46376937677490009712648124896970078050417018260538<br>
74324986199524741059474233309513058123726617309629<br>
919422133635741615725224305... | https://projecteuler.net/problem=13 | 5537376230 |
14 | The following iterative sequence is defined for the set of positive integers:
- $n \to n/2$ ($n$ is even)
- $n \to 3n + 1$ ($n$ is odd)
Using the rule above and starting with $13$, we generate the following sequence:
$$13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1.$$
It can be seen that this sequence... | <p>The following iterative sequence is defined for the set of positive integers:</p>
<ul style="list-style-type:none;">
<li>$n \to n/2$ ($n$ is even)</li>
<li>$n \to 3n + 1$ ($n$ is odd)</li></ul>
<p>Using the rule above and starting with $13$, we generate the following sequence:
$$13 \to 40 \to 20 \to 10 \to 5 \to 16 ... | https://projecteuler.net/problem=14 | 837799 |
15 | Starting in the top left corner of a $2 \times 2$ grid, and only being able to move to the right and down, there are exactly $6$ routes to the bottom right corner.
How many such routes are there through a $20 \times 20$ grid? | <p>Starting in the top left corner of a $2 \times 2$ grid, and only being able to move to the right and down, there are exactly $6$ routes to the bottom right corner.</p>
<div class="center">
<img src="resources/images/0015.png?1678992052" class="dark_img" alt=""></div>
<p>How many such routes are there through a $20 \... | https://projecteuler.net/problem=15 | 137846528820 |
16 | $2^{15} = 32768$ and the sum of its digits is $3 + 2 + 7 + 6 + 8 = 26$.
What is the sum of the digits of the number $2^{1000}$? | <p>$2^{15} = 32768$ and the sum of its digits is $3 + 2 + 7 + 6 + 8 = 26$.</p>
<p>What is the sum of the digits of the number $2^{1000}$?</p> | https://projecteuler.net/problem=16 | 1366 |
17 | If the numbers $1$ to $5$ are written out in words: one, two, three, four, five, then there are $3 + 3 + 5 + 4 + 4 = 19$ letters used in total.
If all the numbers from $1$ to $1000$ (one thousand) inclusive were written out in words, how many letters would be used?
NOTE: Do not count spaces or hyphens. For example, $... | <p>If the numbers $1$ to $5$ are written out in words: one, two, three, four, five, then there are $3 + 3 + 5 + 4 + 4 = 19$ letters used in total.</p>
<p>If all the numbers from $1$ to $1000$ (one thousand) inclusive were written out in words, how many letters would be used? </p>
<br><p class="note"><b>NOTE:</b> Do not... | https://projecteuler.net/problem=17 | 21124 |
18 | By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is $23$.
3
7 4
2 4 6
8 5 9 3
That is, $3 + 7 + 4 + 9 = 23$.
Find the maximum total from top to bottom of the triangle below:
75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23... | <p>By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is $23$.</p>
<p class="monospace center"><span class="red"><b>3</b></span><br><span class="red"><b>7</b></span> 4<br>
2 <span class="red"><b>4</b></span> 6<br>
8 5 <span class="red"><b>9... | https://projecteuler.net/problem=18 | 1074 |
19 | You are given the following information, but you may prefer to do some research for yourself.
- 1 Jan 1900 was a Monday.
- Thirty days has September,
April, June and November.
All the rest have thirty-one,
Saving February alone,
Which has twenty-eight, rain or shine.
And on leap years, twenty-nine.
- A leap yea... | <p>You are given the following information, but you may prefer to do some research for yourself.</p>
<ul><li>1 Jan 1900 was a Monday.</li>
<li>Thirty days has September,<br>
April, June and November.<br>
All the rest have thirty-one,<br>
Saving February alone,<br>
Which has twenty-eight, rain or shine.<br>
And on leap ... | https://projecteuler.net/problem=19 | 171 |
20 | $n!$ means $n \times (n - 1) \times \cdots \times 3 \times 2 \times 1$.
For example, $10! = 10 \times 9 \times \cdots \times 3 \times 2 \times 1 = 3628800$,
and the sum of the digits in the number $10!$ is $3 + 6 + 2 + 8 + 8 + 0 + 0 = 27$.
Find the sum of the digits in the number $100!$. | <p>$n!$ means $n \times (n - 1) \times \cdots \times 3 \times 2 \times 1$.</p>
<p>For example, $10! = 10 \times 9 \times \cdots \times 3 \times 2 \times 1 = 3628800$,<br>and the sum of the digits in the number $10!$ is $3 + 6 + 2 + 8 + 8 + 0 + 0 = 27$.</p>
<p>Find the sum of the digits in the number $100!$.</p> | https://projecteuler.net/problem=20 | 648 |
21 | Let $d(n)$ be defined as the sum of proper divisors of $n$ (numbers less than $n$ which divide evenly into $n$).
If $d(a) = b$ and $d(b) = a$, where $a \ne b$, then $a$ and $b$ are an amicable pair and each of $a$ and $b$ are called amicable numbers.
For example, the proper divisors of $220$ are $1, 2, 4, 5, 10, 11, ... | <p>Let $d(n)$ be defined as the sum of proper divisors of $n$ (numbers less than $n$ which divide evenly into $n$).<br>
If $d(a) = b$ and $d(b) = a$, where $a \ne b$, then $a$ and $b$ are an amicable pair and each of $a$ and $b$ are called amicable numbers.</p>
<p>For example, the proper divisors of $220$ are $1, 2, 4,... | https://projecteuler.net/problem=21 | 31626 |
22 | Using names.txt (right click and 'Save Link/Target As...'), a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for each name, multiply this value by its alphabetical position in the list to obtain a name score.
For example, wh... | <p>Using <a href="resources/documents/0022_names.txt">names.txt</a> (right click and 'Save Link/Target As...'), a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for each name, multiply this value by its alphabetical position ... | https://projecteuler.net/problem=22 | 871198282 |
23 | A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of $28$ would be $1 + 2 + 4 + 7 + 14 = 28$, which means that $28$ is a perfect number.
A number $n$ is called deficient if the sum of its proper divisors is less than $n$ an... | <p>A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of $28$ would be $1 + 2 + 4 + 7 + 14 = 28$, which means that $28$ is a perfect number.</p>
<p>A number $n$ is called deficient if the sum of its proper divisors is less th... | https://projecteuler.net/problem=23 | 4179871 |
24 | A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:
012 021 102 120 201 210
What i... | <p>A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:</p>
<p class="center">012 021 102 ... | https://projecteuler.net/problem=24 | 2783915460 |
25 | The Fibonacci sequence is defined by the recurrence relation:
$F_n = F_{n - 1} + F_{n - 2}$, where $F_1 = 1$ and $F_2 = 1$.
Hence the first $12$ terms will be:
$$\begin{align}
F_1 &= 1\\
F_2 &= 1\\
F_3 &= 2\\
F_4 &= 3\\
F_5 &= 5\\
F_6 &= 8\\
F_7 &= 13\\
F_8 &= 21\\
F_9 &= 34\\
F_{10} &= 55\\
F_{11} &= 89\\
F_{12} &= ... | <p>The Fibonacci sequence is defined by the recurrence relation:</p>
<blockquote>$F_n = F_{n - 1} + F_{n - 2}$, where $F_1 = 1$ and $F_2 = 1$.</blockquote>
<p>Hence the first $12$ terms will be:</p>
$$\begin{align}
F_1 &= 1\\
F_2 &= 1\\
F_3 &= 2\\
F_4 &= 3\\
F_5 &= 5\\
F_6 &= 8\\
F_7 &= 13\\... | https://projecteuler.net/problem=25 | 4782 |
26 | A unit fraction contains $1$ in the numerator. The decimal representation of the unit fractions with denominators $2$ to $10$ are given:
$$\begin{align}
1/2 &= 0.5\\
1/3 &=0.(3)\\
1/4 &=0.25\\
1/5 &= 0.2\\
1/6 &= 0.1(6)\\
1/7 &= 0.(142857)\\
1/8 &= 0.125\\
1/9 &= 0.(1)\\
1/10 &= 0.1
\end{align}$$
Where $0.1(6)$ means ... | <p>A unit fraction contains $1$ in the numerator. The decimal representation of the unit fractions with denominators $2$ to $10$ are given:</p>
$$\begin{align}
1/2 &= 0.5\\
1/3 &=0.(3)\\
1/4 &=0.25\\
1/5 &= 0.2\\
1/6 &= 0.1(6)\\
1/7 &= 0.(142857)\\
1/8 &= 0.125\\
1/9 &= 0.(1)\\
1/10 &... | https://projecteuler.net/problem=26 | 983 |
27 | Euler discovered the remarkable quadratic formula:
$n^2 + n + 41$
It turns out that the formula will produce $40$ primes for the consecutive integer values $0 \le n \le 39$. However, when $n = 40, 40^2 + 40 + 41 = 40(40 + 1) + 41$ is divisible by $41$, and certainly when $n = 41, 41^2 + 41 + 41$ is clearly divisible ... | <p>Euler discovered the remarkable quadratic formula:</p>
<p class="center">$n^2 + n + 41$</p>
<p>It turns out that the formula will produce $40$ primes for the consecutive integer values $0 \le n \le 39$. However, when $n = 40, 40^2 + 40 + 41 = 40(40 + 1) + 41$ is divisible by $41$, and certainly when $n = 41, 41^2 + ... | https://projecteuler.net/problem=27 | -59231 |
28 | Starting with the number $1$ and moving to the right in a clockwise direction a $5$ by $5$ spiral is formed as follows:
21 22 23 24 25
20 7 8 9 10
19 6 1 2 11
18 5 4 3 12
17 16 15 14 13
It can be verified that the sum of the numbers on the diagonals is $101$.
What is the sum of the numbers on the diagona... | <p>Starting with the number $1$ and moving to the right in a clockwise direction a $5$ by $5$ spiral is formed as follows:</p>
<p class="monospace center"><span class="red"><b>21</b></span> 22 23 24 <span class="red"><b>25</b></span><br>
20 <span class="red"><b>7</b></span> 8 <span class="red"><b>9</b></span> 10<br>... | https://projecteuler.net/problem=28 | 669171001 |
29 | Consider all integer combinations of $a^b$ for $2 \le a \le 5$ and $2 \le b \le 5$:
$$\begin{array}{rrrr}
2^2=4, &2^3=8, &2^4=16, &2^5=32\\
3^2=9, &3^3=27, &3^4=81, &3^5=243\\
4^2=16, &4^3=64, &4^4=256, &4^5=1024\\
5^2=25, &5^3=125, &5^4=625, &5^5=3125
\end{array}$$
If they are then placed in numerical order, with any... | <p>Consider all integer combinations of $a^b$ for $2 \le a \le 5$ and $2 \le b \le 5$:
$$\begin{array}{rrrr}
2^2=4, &2^3=8, &2^4=16, &2^5=32\\
3^2=9, &3^3=27, &3^4=81, &3^5=243\\
4^2=16, &4^3=64, &4^4=256, &4^5=1024\\
5^2=25, &5^3=125, &5^4=625, &5^5=3125
\end{array}$$</p... | https://projecteuler.net/problem=29 | 9183 |
30 | Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:
$$\begin{align}
1634 &= 1^4 + 6^4 + 3^4 + 4^4\\
8208 &= 8^4 + 2^4 + 0^4 + 8^4\\
9474 &= 9^4 + 4^4 + 7^4 + 4^4
\end{align}$$
As $1 = 1^4$ is not a sum it is not included.
The sum of these numbers is $1634 + 8208 ... | <p>Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:
$$\begin{align}
1634 &= 1^4 + 6^4 + 3^4 + 4^4\\
8208 &= 8^4 + 2^4 + 0^4 + 8^4\\
9474 &= 9^4 + 4^4 + 7^4 + 4^4
\end{align}$$
</p><p class="smaller">As $1 = 1^4$ is not a sum it is not included.</p>
<... | https://projecteuler.net/problem=30 | 443839 |
31 | In the United Kingdom the currency is made up of pound (£) and pence (p). There are eight coins in general circulation:
1p, 2p, 5p, 10p, 20p, 50p, £1 (100p), and £2 (200p).
It is possible to make £2 in the following way:
1×£1 + 1×50p + 2×20p + 1×5p + 1×2p + 3×1p
How many different ways can £2 be made using any number... | <p>In the United Kingdom the currency is made up of pound (£) and pence (p). There are eight coins in general circulation:</p>
<blockquote>1p, 2p, 5p, 10p, 20p, 50p, £1 (100p), and £2 (200p).</blockquote>
<p>It is possible to make £2 in the following way:</p>
<blockquote>1×£1 + 1×50p + 2×20p + 1×5p + 1×2p + 3×1p</block... | https://projecteuler.net/problem=31 | 73682 |
32 | We shall say that an $n$-digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once; for example, the $5$-digit number, $15234$, is $1$ through $5$ pandigital.
The product $7254$ is unusual, as the identity, $39 \times 186 = 7254$, containing multiplicand, multiplier, and product is $1$ throu... | <p>We shall say that an $n$-digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once; for example, the $5$-digit number, $15234$, is $1$ through $5$ pandigital.</p>
<p>The product $7254$ is unusual, as the identity, $39 \times 186 = 7254$, containing multiplicand, multiplier, and product is... | https://projecteuler.net/problem=32 | 45228 |
33 | The fraction $49/98$ is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that $49/98 = 4/8$, which is correct, is obtained by cancelling the $9$s.
We shall consider fractions like, $30/50 = 3/5$, to be trivial examples.
There are exactly four non-trivial examp... | <p>The fraction $49/98$ is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that $49/98 = 4/8$, which is correct, is obtained by cancelling the $9$s.</p>
<p>We shall consider fractions like, $30/50 = 3/5$, to be trivial examples.</p>
<p>There are exactly four no... | https://projecteuler.net/problem=33 | 100 |
34 | $145$ is a curious number, as $1! + 4! + 5! = 1 + 24 + 120 = 145$.
Find the sum of all numbers which are equal to the sum of the factorial of their digits.
Note: As $1! = 1$ and $2! = 2$ are not sums they are not included. | <p>$145$ is a curious number, as $1! + 4! + 5! = 1 + 24 + 120 = 145$.</p>
<p>Find the sum of all numbers which are equal to the sum of the factorial of their digits.</p>
<p class="smaller">Note: As $1! = 1$ and $2! = 2$ are not sums they are not included.</p> | https://projecteuler.net/problem=34 | 40730 |
35 | The number, $197$, is called a circular prime because all rotations of the digits: $197$, $971$, and $719$, are themselves prime.
There are thirteen such primes below $100$: $2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79$, and $97$.
How many circular primes are there below one million? | <p>The number, $197$, is called a circular prime because all rotations of the digits: $197$, $971$, and $719$, are themselves prime.</p>
<p>There are thirteen such primes below $100$: $2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79$, and $97$.</p>
<p>How many circular primes are there below one million?</p> | https://projecteuler.net/problem=35 | 55 |
36 | The decimal number, $585 = 1001001001_2$ (binary), is palindromic in both bases.
Find the sum of all numbers, less than one million, which are palindromic in base $10$ and base $2$.
(Please note that the palindromic number, in either base, may not include leading zeros.) | <p>The decimal number, $585 = 1001001001_2$ (binary), is palindromic in both bases.</p>
<p>Find the sum of all numbers, less than one million, which are palindromic in base $10$ and base $2$.</p>
<p class="smaller">(Please note that the palindromic number, in either base, may not include leading zeros.)</p> | https://projecteuler.net/problem=36 | 872187 |
37 | The number $3797$ has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: $3797$, $797$, $97$, and $7$. Similarly we can work from right to left: $3797$, $379$, $37$, and $3$.
Find the sum of the only eleven primes that are both ... | <p>The number $3797$ has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: $3797$, $797$, $97$, and $7$. Similarly we can work from right to left: $3797$, $379$, $37$, and $3$.</p>
<p>Find the sum of the only eleven primes that ... | https://projecteuler.net/problem=37 | 748317 |
38 | Take the number $192$ and multiply it by each of $1$, $2$, and $3$:
$$\begin{align}
192 \times 1 &= 192\\
192 \times 2 &= 384\\
192 \times 3 &= 576
\end{align}$$
By concatenating each product we get the $1$ to $9$ pandigital, $192384576$. We will call $192384576$ the concatenated product of $192$ and $(1,2,3)$.
The s... | <p>Take the number $192$ and multiply it by each of $1$, $2$, and $3$:</p>
$$\begin{align}
192 \times 1 &= 192\\
192 \times 2 &= 384\\
192 \times 3 &= 576
\end{align}$$
<p>By concatenating each product we get the $1$ to $9$ pandigital, $192384576$. We will call $192384576$ the concatenated product of $192$ ... | https://projecteuler.net/problem=38 | 932718654 |
39 | If $p$ is the perimeter of a right angle triangle with integral length sides, $\{a, b, c\}$, there are exactly three solutions for $p = 120$.
$\{20,48,52\}$, $\{24,45,51\}$, $\{30,40,50\}$
For which value of $p \le 1000$, is the number of solutions maximised? | <p>If $p$ is the perimeter of a right angle triangle with integral length sides, $\{a, b, c\}$, there are exactly three solutions for $p = 120$.</p>
<p>$\{20,48,52\}$, $\{24,45,51\}$, $\{30,40,50\}$</p>
<p>For which value of $p \le 1000$, is the number of solutions maximised?</p> | https://projecteuler.net/problem=39 | 840 |
40 | An irrational decimal fraction is created by concatenating the positive integers:
$$0.12345678910{\color{red}\mathbf 1}112131415161718192021\cdots$$
It can be seen that the $12$th digit of the fractional part is $1$.
If $d_n$ represents the $n$th digit of the fractional part, find the value of the following expressio... | <p>An irrational decimal fraction is created by concatenating the positive integers:
$$0.12345678910{\color{red}\mathbf 1}112131415161718192021\cdots$$</p>
<p>It can be seen that the $12$<sup>th</sup> digit of the fractional part is $1$.</p>
<p>If $d_n$ represents the $n$<sup>th</sup> digit of the fractional part, find... | https://projecteuler.net/problem=40 | 210 |
41 | We shall say that an $n$-digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once. For example, $2143$ is a $4$-digit pandigital and is also prime.
What is the largest $n$-digit pandigital prime that exists? | <p>We shall say that an $n$-digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once. For example, $2143$ is a $4$-digit pandigital and is also prime.</p>
<p>What is the largest $n$-digit pandigital prime that exists?</p> | https://projecteuler.net/problem=41 | 7652413 |
42 | The $n$th term of the sequence of triangle numbers is given by, $t_n = \frac12n(n+1)$; so the first ten triangle numbers are:
$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$$
By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For examp... | <p>The $n$<sup>th</sup> term of the sequence of triangle numbers is given by, $t_n = \frac12n(n+1)$; so the first ten triangle numbers are:
$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$$</p>
<p>By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a w... | https://projecteuler.net/problem=42 | 162 |
43 | The number, $1406357289$, is a $0$ to $9$ pandigital number because it is made up of each of the digits $0$ to $9$ in some order, but it also has a rather interesting sub-string divisibility property.
Let $d_1$ be the $1$st digit, $d_2$ be the $2$nd digit, and so on. In this way, we note the following:
- $d_2d_3d_4=4... | <p>The number, $1406357289$, is a $0$ to $9$ pandigital number because it is made up of each of the digits $0$ to $9$ in some order, but it also has a rather interesting sub-string divisibility property.</p>
<p>Let $d_1$ be the $1$<sup>st</sup> digit, $d_2$ be the $2$<sup>nd</sup> digit, and so on. In this way, we note... | https://projecteuler.net/problem=43 | 16695334890 |
44 | Pentagonal numbers are generated by the formula, $P_n=n(3n-1)/2$. The first ten pentagonal numbers are:
$$1, 5, 12, 22, 35, 51, 70, 92, 117, 145, \dots$$
It can be seen that $P_4 + P_7 = 22 + 70 = 92 = P_8$. However, their difference, $70 - 22 = 48$, is not pentagonal.
Find the pair of pentagonal numbers, $P_j$ and $... | <p>Pentagonal numbers are generated by the formula, $P_n=n(3n-1)/2$. The first ten pentagonal numbers are:
$$1, 5, 12, 22, 35, 51, 70, 92, 117, 145, \dots$$</p>
<p>It can be seen that $P_4 + P_7 = 22 + 70 = 92 = P_8$. However, their difference, $70 - 22 = 48$, is not pentagonal.</p>
<p>Find the pair of pentagonal numbe... | https://projecteuler.net/problem=44 | 5482660 |
45 | Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
| Triangle | | $T_n=n(n+1)/2$ | | $1, 3, 6, 10, 15, \dots$ |
| Pentagonal | | $P_n=n(3n - 1)/2$ | | $1, 5, 12, 22, 35, \dots$ |
| Hexagonal | | $H_n=n(2n - 1)$ | | $1, 6, 15, 28, 45, \dots$ |
It can be verified that $T_{285} = ... | <p>Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:</p>
<table><tr><td>Triangle</td>
<td> </td>
<td>$T_n=n(n+1)/2$</td>
<td> </td>
<td>$1, 3, 6, 10, 15, \dots$</td>
</tr><tr><td>Pentagonal</td>
<td> </td>
<td>$P_n=n(3n - 1)/2$</td>
<td> </td>
<td>$1, 5, 12, 22, 35, \dots$</td>
</tr><... | https://projecteuler.net/problem=45 | 1533776805 |
46 | It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.
$$\begin{align}
9 = 7 + 2 \times 1^2\\
15 = 7 + 2 \times 2^2\\
21 = 3 + 2 \times 3^2\\
25 = 7 + 2 \times 3^2\\
27 = 19 + 2 \times 2^2\\
33 = 31 + 2 \times 1^2
\end{align}$$
It turns out that t... | <p>It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.</p>
$$\begin{align}
9 = 7 + 2 \times 1^2\\
15 = 7 + 2 \times 2^2\\
21 = 3 + 2 \times 3^2\\
25 = 7 + 2 \times 3^2\\
27 = 19 + 2 \times 2^2\\
33 = 31 + 2 \times 1^2
\end{align}$$
<p>It turns o... | https://projecteuler.net/problem=46 | 5777 |
47 | The first two consecutive numbers to have two distinct prime factors are:
$$\begin{align}
14 &= 2 \times 7\\
15 &= 3 \times 5.
\end{align}$$
The first three consecutive numbers to have three distinct prime factors are:
$$\begin{align}
644 &= 2^2 \times 7 \times 23\\
645 &= 3 \times 5 \times 43\\
646 &= 2 \times 17 \t... | <p>The first two consecutive numbers to have two distinct prime factors are:</p>
$$\begin{align}
14 &= 2 \times 7\\
15 &= 3 \times 5.
\end{align}$$
<p>The first three consecutive numbers to have three distinct prime factors are:</p>
$$\begin{align}
644 &= 2^2 \times 7 \times 23\\
645 &= 3 \times 5 \time... | https://projecteuler.net/problem=47 | 134043 |
48 | The series, $1^1 + 2^2 + 3^3 + \cdots + 10^{10} = 10405071317$.
Find the last ten digits of the series, $1^1 + 2^2 + 3^3 + \cdots + 1000^{1000}$. | <p>The series, $1^1 + 2^2 + 3^3 + \cdots + 10^{10} = 10405071317$.</p>
<p>Find the last ten digits of the series, $1^1 + 2^2 + 3^3 + \cdots + 1000^{1000}$.</p> | https://projecteuler.net/problem=48 | 9110846700 |
49 | The arithmetic sequence, $1487, 4817, 8147$, in which each of the terms increases by $3330$, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the $4$-digit numbers are permutations of one another.
There are no arithmetic sequences made up of three $1$-, $2$-, or $3$-digit primes, exhibi... | <p>The arithmetic sequence, $1487, 4817, 8147$, in which each of the terms increases by $3330$, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the $4$-digit numbers are permutations of one another.</p>
<p>There are no arithmetic sequences made up of three $1$-, $2$-, or $3$-digit prime... | https://projecteuler.net/problem=49 | 296962999629 |
50 | The prime $41$, can be written as the sum of six consecutive primes:
$$41 = 2 + 3 + 5 + 7 + 11 + 13.$$
This is the longest sum of consecutive primes that adds to a prime below one-hundred.
The longest sum of consecutive primes below one-thousand that adds to a prime, contains $21$ terms, and is equal to $953$.
Which... | <p>The prime $41$, can be written as the sum of six consecutive primes:</p>
$$41 = 2 + 3 + 5 + 7 + 11 + 13.$$
<p>This is the longest sum of consecutive primes that adds to a prime below one-hundred.</p>
<p>The longest sum of consecutive primes below one-thousand that adds to a prime, contains $21$ terms, and is equal t... | https://projecteuler.net/problem=50 | 997651 |
51 | By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.
By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes among the ten generated numbers, yielding the... | <p>By replacing the 1<sup>st</sup> digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.</p>
<p>By replacing the 3<sup>rd</sup> and 4<sup>th</sup> digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes amo... | https://projecteuler.net/problem=51 | 121313 |
52 | It can be seen that the number, $125874$, and its double, $251748$, contain exactly the same digits, but in a different order.
Find the smallest positive integer, $x$, such that $2x$, $3x$, $4x$, $5x$, and $6x$, contain the same digits. | <p>It can be seen that the number, $125874$, and its double, $251748$, contain exactly the same digits, but in a different order.</p>
<p>Find the smallest positive integer, $x$, such that $2x$, $3x$, $4x$, $5x$, and $6x$, contain the same digits.</p> | https://projecteuler.net/problem=52 | 142857 |
53 | There are exactly ten ways of selecting three from five, 12345:
123, 124, 125, 134, 135, 145, 234, 235, 245, and 345
In combinatorics, we use the notation, $\displaystyle \binom 5 3 = 10$.
In general, $\displaystyle \binom n r = \dfrac{n!}{r!(n-r)!}$, where $r \le n$, $n! = n \times (n-1) \times ... \times 3 \times ... | <p>There are exactly ten ways of selecting three from five, 12345:</p>
<p class="center">123, 124, 125, 134, 135, 145, 234, 235, 245, and 345</p>
<p>In combinatorics, we use the notation, $\displaystyle \binom 5 3 = 10$.</p>
<p>In general, $\displaystyle \binom n r = \dfrac{n!}{r!(n-r)!}$, where $r \le n$, $n! = n \tim... | https://projecteuler.net/problem=53 | 4075 |
54 | In the card game poker, a hand consists of five cards and are ranked, from lowest to highest, in the following way:
- High Card: Highest value card.
- One Pair: Two cards of the same value.
- Two Pairs: Two different pairs.
- Three of a Kind: Three cards of the same value.
- Straight: All cards are consecutive val... | <p>In the card game poker, a hand consists of five cards and are ranked, from lowest to highest, in the following way:</p>
<ul><li><b>High Card</b>: Highest value card.</li>
<li><b>One Pair</b>: Two cards of the same value.</li>
<li><b>Two Pairs</b>: Two different pairs.</li>
<li><b>Three of a Kind</b>: Three cards of ... | https://projecteuler.net/problem=54 | 376 |
55 | If we take $47$, reverse and add, $47 + 74 = 121$, which is palindromic.
Not all numbers produce palindromes so quickly. For example,
$$\begin{align}
349 + 943 &= 1292\\
1292 + 2921 &= 4213\\
4213 + 3124 &= 7337
\end{align}$$
That is, $349$ took three iterations to arrive at a palindrome.
Although no one has proved ... | <p>If we take $47$, reverse and add, $47 + 74 = 121$, which is palindromic.</p>
<p>Not all numbers produce palindromes so quickly. For example,</p>
$$\begin{align}
349 + 943 &= 1292\\
1292 + 2921 &= 4213\\
4213 + 3124 &= 7337
\end{align}$$
<p>That is, $349$ took three iterations to arrive at a palindrome.</... | https://projecteuler.net/problem=55 | 249 |
56 | A googol ($10^{100}$) is a massive number: one followed by one-hundred zeros; $100^{100}$ is almost unimaginably large: one followed by two-hundred zeros. Despite their size, the sum of the digits in each number is only $1$.
Considering natural numbers of the form, $a^b$, where $a, b \lt 100$, what is the maximum digi... | <p>A googol ($10^{100}$) is a massive number: one followed by one-hundred zeros; $100^{100}$ is almost unimaginably large: one followed by two-hundred zeros. Despite their size, the sum of the digits in each number is only $1$.</p>
<p>Considering natural numbers of the form, $a^b$, where $a, b \lt 100$, what is the max... | https://projecteuler.net/problem=56 | 972 |
57 | It is possible to show that the square root of two can be expressed as an infinite continued fraction.
$\sqrt 2 =1+ \frac 1 {2+ \frac 1 {2 +\frac 1 {2+ \dots}}}$
By expanding this for the first four iterations, we get:
$1 + \frac 1 2 = \frac 32 = 1.5$
$1 + \frac 1 {2 + \frac 1 2} = \frac 7 5 = 1.4$
$1 + \frac 1 {... | <p>It is possible to show that the square root of two can be expressed as an infinite continued fraction.</p>
<p class="center">$\sqrt 2 =1+ \frac 1 {2+ \frac 1 {2 +\frac 1 {2+ \dots}}}$</p>
<p>By expanding this for the first four iterations, we get:</p>
<p>$1 + \frac 1 2 = \frac 32 = 1.5$<br>
$1 + \frac 1 {2 + \frac ... | https://projecteuler.net/problem=57 | 153 |
58 | Starting with $1$ and spiralling anticlockwise in the following way, a square spiral with side length $7$ is formed.
37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18 5 4 3 12 29
40 19 6 1 2 11 28
41 20 7 8 9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49
It is interesting to note that the odd squares li... | <p>Starting with $1$ and spiralling anticlockwise in the following way, a square spiral with side length $7$ is formed.</p>
<p class="center monospace"><span class="red"><b>37</b></span> 36 35 34 33 32 <span class="red"><b>31</b></span><br>
38 <span class="red"><b>17</b></span> 16 15 14 <span class="red"><b>13</b></spa... | https://projecteuler.net/problem=58 | 26241 |
59 | Each character on a computer is assigned a unique code and the preferred standard is ASCII (American Standard Code for Information Interchange). For example, uppercase A = 65, asterisk (*) = 42, and lowercase k = 107.
A modern encryption method is to take a text file, convert the bytes to ASCII, then XOR each byte wit... | <p>Each character on a computer is assigned a unique code and the preferred standard is ASCII (American Standard Code for Information Interchange). For example, uppercase A = 65, asterisk (*) = 42, and lowercase k = 107.</p>
<p>A modern encryption method is to take a text file, convert the bytes to ASCII, then XOR each... | https://projecteuler.net/problem=59 | 129448 |
60 | The primes $3$, $7$, $109$, and $673$, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking $7$ and $109$, both $7109$ and $1097$ are prime. The sum of these four primes, $792$, represents the lowest sum for a set of four primes with thi... | <p>The primes $3$, $7$, $109$, and $673$, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking $7$ and $109$, both $7109$ and $1097$ are prime. The sum of these four primes, $792$, represents the lowest sum for a set of four primes with ... | https://projecteuler.net/problem=60 | 26033 |
61 | Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:
| Triangle | | $P_{3,n}=n(n+1)/2$ | | $1, 3, 6, 10, 15, \dots$ |
| Square | | $P_{4,n}=n^2$ | | $1, 4, 9, 16, 25, \dots$ |
| Pentagonal | | $P_{5,n}=n(3n-1)/... | <p>Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:</p>
<table><tr><td>Triangle</td>
<td> </td>
<td>$P_{3,n}=n(n+1)/2$</td>
<td> </td>
<td>$1, 3, 6, 10, 15, \dots$</td>
</tr><tr><td>Square</td>
<td> </td>
<td>$P_... | https://projecteuler.net/problem=61 | 28684 |
62 | The cube, $41063625$ ($345^3$), can be permuted to produce two other cubes: $56623104$ ($384^3$) and $66430125$ ($405^3$). In fact, $41063625$ is the smallest cube which has exactly three permutations of its digits which are also cube.
Find the smallest cube for which exactly five permutations of its digits are cube. | <p>The cube, $41063625$ ($345^3$), can be permuted to produce two other cubes: $56623104$ ($384^3$) and $66430125$ ($405^3$). In fact, $41063625$ is the smallest cube which has exactly three permutations of its digits which are also cube.</p>
<p>Find the smallest cube for which exactly five permutations of its digits a... | https://projecteuler.net/problem=62 | 127035954683 |
63 | The $5$-digit number, $16807=7^5$, is also a fifth power. Similarly, the $9$-digit number, $134217728=8^9$, is a ninth power.
How many $n$-digit positive integers exist which are also an $n$th power? | <p>The $5$-digit number, $16807=7^5$, is also a fifth power. Similarly, the $9$-digit number, $134217728=8^9$, is a ninth power.</p>
<p>How many $n$-digit positive integers exist which are also an $n$th power?</p> | https://projecteuler.net/problem=63 | 49 |
64 | All square roots are periodic when written as continued fractions and can be written in the form:
$$\sqrt{N}=a_0 + \dfrac 1 {a_1 + \dfrac 1 {a_2 + \dfrac 1 {a_3 + \dots}}}$$
For example, let us consider $\sqrt{23}:$
$$\sqrt{23} = 4 + \sqrt{23}-4=4 + \dfrac 1 {\dfrac 1 {\sqrt{23}-4}} = 4+\dfrac 1 {1 + \dfrac{\sqrt{2... | <p>All square roots are periodic when written as continued fractions and can be written in the form:</p>
$$\sqrt{N}=a_0 + \dfrac 1 {a_1 + \dfrac 1 {a_2 + \dfrac 1 {a_3 + \dots}}}$$
<p>For example, let us consider $\sqrt{23}:$</p>
$$\sqrt{23} = 4 + \sqrt{23}-4=4 + \dfrac 1 {\dfrac 1 {\sqrt{23}-4}} = 4+\dfrac 1 {1 + \... | https://projecteuler.net/problem=64 | 1322 |
65 | The square root of $2$ can be written as an infinite continued fraction.
$$\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + ...}}}}$$
The infinite continued fraction can be written, $\sqrt{2} = [1; (2)]$, $(2)$ indicates that $2$ repeats ad infinitum. In a similar way, $\sqrt{23} = [4; (1, 3, 1,... | <p>The square root of $2$ can be written as an infinite continued fraction.</p>
<p>$$\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + ...}}}}$$</p>
<p>The infinite continued fraction can be written, $\sqrt{2} = [1; (2)]$, $(2)$ indicates that $2$ repeats <i>ad infinitum</i>. In a similar way, $\sq... | https://projecteuler.net/problem=65 | 272 |
66 | Consider quadratic Diophantine equations of the form:
$$x^2 - Dy^2 = 1$$
For example, when $D=13$, the minimal solution in $x$ is $649^2 - 13 \times 180^2 = 1$.
It can be assumed that there are no solutions in positive integers when $D$ is square.
By finding minimal solutions in $x$ for $D = \{2, 3, 5, 6, 7\}$, we o... | <p>Consider quadratic Diophantine equations of the form:
$$x^2 - Dy^2 = 1$$</p>
<p>For example, when $D=13$, the minimal solution in $x$ is $649^2 - 13 \times 180^2 = 1$.</p>
<p>It can be assumed that there are no solutions in positive integers when $D$ is square.</p>
<p>By finding minimal solutions in $x$ for $D = \{2... | https://projecteuler.net/problem=66 | 661 |
67 | By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.
3
7 4
2 4 6
8 5 9 3
That is, 3 + 7 + 4 + 9 = 23.
Find the maximum total from top to bottom in triangle.txt (right click and 'Save Link/Target As...'), a 15K text file containing... | <p>By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.</p>
<p class="monospace center"><span class="red"><b>3</b></span><br><span class="red"><b>7</b></span> 4<br>
2 <span class="red"><b>4</b></span> 6<br>
8 5 <span class="red"><b>9</... | https://projecteuler.net/problem=67 | 7273 |
68 | Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.
Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely. For example, the above solution can be described b... | <p>Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.</p>
<div class="center">
<img src="resources/images/0068_1.png?1678992052" class="dark_img" alt=""><br></div>
<p>Working <b>clockwise</b>, and starting from the group of three with the numerically lowest external... | https://projecteuler.net/problem=68 | 6531031914842725 |
69 | Euler's totient function, $\phi(n)$ [sometimes called the phi function], is defined as the number of positive integers not exceeding $n$ which are relatively prime to $n$. For example, as $1$, $2$, $4$, $5$, $7$, and $8$, are all less than or equal to nine and relatively prime to nine, $\phi(9)=6$.
| $n$ | Relatively ... | <p>Euler's totient function, $\phi(n)$ [sometimes called the phi function], is defined as the number of positive integers not exceeding $n$ which are relatively prime to $n$. For example, as $1$, $2$, $4$, $5$, $7$, and $8$, are all less than or equal to nine and relatively prime to nine, $\phi(9)=6$.</p>
<div class="c... | https://projecteuler.net/problem=69 | 510510 |
70 | Euler's totient function, $\phi(n)$ [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to $n$ which are relatively prime to $n$. For example, as $1, 2, 4, 5, 7$, and $8$, are all less than nine and relatively prime to nine, $\phi(9)=6$.
The number $1$ is consider... | <p>Euler's totient function, $\phi(n)$ [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to $n$ which are relatively prime to $n$. For example, as $1, 2, 4, 5, 7$, and $8$, are all less than nine and relatively prime to nine, $\phi(9)=6$.<br>The number $1$ is co... | https://projecteuler.net/problem=70 | 8319823 |
71 | Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \frac 1 4, \... | <p>Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction.</p>
<p>If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \fr... | https://projecteuler.net/problem=71 | 428570 |
72 | Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \frac 1 4, \... | <p>Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction.</p>
<p>If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \fr... | https://projecteuler.net/problem=72 | 303963552391 |
73 | Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n, d)=1$, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \frac 1 4, ... | <p>Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n, d)=1$, it is called a reduced proper fraction.</p>
<p>If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \f... | https://projecteuler.net/problem=73 | 7295372 |
74 | The number $145$ is well known for the property that the sum of the factorial of its digits is equal to $145$:
$$1! + 4! + 5! = 1 + 24 + 120 = 145.$$
Perhaps less well known is $169$, in that it produces the longest chain of numbers that link back to $169$; it turns out that there are only three such loops that exist:... | <p>The number $145$ is well known for the property that the sum of the factorial of its digits is equal to $145$:
$$1! + 4! + 5! = 1 + 24 + 120 = 145.$$</p>
<p>Perhaps less well known is $169$, in that it produces the longest chain of numbers that link back to $169$; it turns out that there are only three such loops th... | https://projecteuler.net/problem=74 | 402 |
75 | It turns out that $\pu{12 cm}$ is the smallest length of wire that can be bent to form an integer sided right angle triangle in exactly one way, but there are many more examples.
- $\pu{\mathbf{12} \mathbf{cm}}$: $(3,4,5)$
- $\pu{\mathbf{24} \mathbf{cm}}$: $(6,8,10)$
- $\pu{\mathbf{30} \mathbf{cm}}$: $(5,12,13)$
- ... | <p>It turns out that $\pu{12 cm}$ is the smallest length of wire that can be bent to form an integer sided right angle triangle in exactly one way, but there are many more examples.</p>
<ul style="list-style-type:none;">
<li>$\pu{\mathbf{12} \mathbf{cm}}$: $(3,4,5)$</li>
<li>$\pu{\mathbf{24} \mathbf{cm}}$: $(6,8,10)$</... | https://projecteuler.net/problem=75 | 161667 |
76 | It is possible to write five as a sum in exactly six different ways:
$$\begin{align}
&4 + 1\\
&3 + 2\\
&3 + 1 + 1\\
&2 + 2 + 1\\
&2 + 1 + 1 + 1\\
&1 + 1 + 1 + 1 + 1
\end{align}$$
How many different ways can one hundred be written as a sum of at least two positive integers? | <p>It is possible to write five as a sum in exactly six different ways:</p>
$$\begin{align}
&4 + 1\\
&3 + 2\\
&3 + 1 + 1\\
&2 + 2 + 1\\
&2 + 1 + 1 + 1\\
&1 + 1 + 1 + 1 + 1
\end{align}$$
<p>How many different ways can one hundred be written as a sum of at least two positive integers?</p> | https://projecteuler.net/problem=76 | 190569291 |
77 | It is possible to write ten as the sum of primes in exactly five different ways:
$$\begin{align}
&7 + 3\\
&5 + 5\\
&5 + 3 + 2\\
&3 + 3 + 2 + 2\\
&2 + 2 + 2 + 2 + 2
\end{align}$$
What is the first value which can be written as the sum of primes in over five thousand different ways? | <p>It is possible to write ten as the sum of primes in exactly five different ways:</p>
$$\begin{align}
&7 + 3\\
&5 + 5\\
&5 + 3 + 2\\
&3 + 3 + 2 + 2\\
&2 + 2 + 2 + 2 + 2
\end{align}$$
<p>What is the first value which can be written as the sum of primes in over five thousand different ways?</p> | https://projecteuler.net/problem=77 | 71 |
78 | Let $p(n)$ represent the number of different ways in which $n$ coins can be separated into piles. For example, five coins can be separated into piles in exactly seven different ways, so $p(5)=7$.
OOOOO
OOOO O
OOO OO
OOO O O
OO OO O
OO O O O
O O O O O
Find the least value of $n$ for whi... | <p>Let $p(n)$ represent the number of different ways in which $n$ coins can be separated into piles. For example, five coins can be separated into piles in exactly seven different ways, so $p(5)=7$.</p>
<div class="margin_left">
OOOOO<br>
OOOO O<br>
OOO OO<br>
OOO O O<br>
OO OO O<br>
OO O O O<br>
O ... | https://projecteuler.net/problem=78 | 55374 |
79 | A common security method used for online banking is to ask the user for three random characters from a passcode. For example, if the passcode was 531278, they may ask for the 2nd, 3rd, and 5th characters; the expected reply would be: 317.
The text file, keylog.txt, contains fifty successful login attempts.
Given that... | <p>A common security method used for online banking is to ask the user for three random characters from a passcode. For example, if the passcode was 531278, they may ask for the 2nd, 3rd, and 5th characters; the expected reply would be: 317.</p>
<p>The text file, <a href="resources/documents/0079_keylog.txt">keylog.txt... | https://projecteuler.net/problem=79 | 73162890 |
80 | It is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is infinite without any repeating pattern at all.
The square root of two is $1.41421356237309504880\cdots$, and the digital sum of the first one hundred decimal digits is $4... | <p>It is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is infinite without any repeating pattern at all.</p>
<p>The square root of two is $1.41421356237309504880\cdots$, and the digital sum of the first one hundred decimal dig... | https://projecteuler.net/problem=80 | 40886 |
81 | In the $5$ by $5$ matrix below, the minimal path sum from the top left to the bottom right, by only moving to the right and down, is indicated in bold red and is equal to $2427$.
$$
\begin{pmatrix}
\color{red}{131} & 673 & 234 & 103 & 18\\
\color{red}{201} & \color{red}{96} & \color{red}{342} & 965 & 150\\
630 & 803 &... | <p>In the $5$ by $5$ matrix below, the minimal path sum from the top left to the bottom right, by <b>only moving to the right and down</b>, is indicated in bold red and is equal to $2427$.</p>
<div class="center">
$$
\begin{pmatrix}
\color{red}{131} & 673 & 234 & 103 & 18\\
\color{red}{201} & \color... | https://projecteuler.net/problem=81 | 427337 |
82 | NOTE: This problem is a more challenging version of Problem 81.
The minimal path sum in the $5$ by $5$ matrix below, by starting in any cell in the left column and finishing in any cell in the right column, and only moving up, down, and right, is indicated in red and bold; the sum is equal to $994$.
$$
\begin{pmatrix... | <p class="small_notice">NOTE: This problem is a more challenging version of <a href="problem=81">Problem 81</a>.</p>
<p>The minimal path sum in the $5$ by $5$ matrix below, by starting in any cell in the left column and finishing in any cell in the right column, and only moving up, down, and right, is indicated in red ... | https://projecteuler.net/problem=82 | 260324 |
83 | NOTE: This problem is a significantly more challenging version of Problem 81.
In the $5$ by $5$ matrix below, the minimal path sum from the top left to the bottom right, by moving left, right, up, and down, is indicated in bold red and is equal to $2297$.
$$
\begin{pmatrix}
\color{red}{131} & 673 & \color{red}{234} &... | <p class="small_notice">NOTE: This problem is a significantly more challenging version of <a href="problem=81">Problem 81</a>.</p>
<p>In the $5$ by $5$ matrix below, the minimal path sum from the top left to the bottom right, by moving left, right, up, and down, is indicated in bold red and is equal to $2297$.</p>
<div... | https://projecteuler.net/problem=83 | 425185 |
84 | In the game, Monopoly, the standard board is set up in the following way:
A player starts on the GO square and adds the scores on two 6-sided dice to determine the number of squares they advance in a clockwise direction. Without any further rules we would expect to visit each square with equal probability: 2.5%. Howev... | <p>In the game, <strong>Monopoly</strong>, the standard board is set up in the following way:</p>
<div class="center">
<img src="resources/images/0084_monopoly_board.png?1678992052" alt="0084_monopoly_board.png">
</div>
<p>A player starts on the GO square and adds the scores on two 6-sided dice to determine the number ... | https://projecteuler.net/problem=84 | 101524 |
85 | By counting carefully it can be seen that a rectangular grid measuring $3$ by $2$ contains eighteen rectangles:
Although there exists no rectangular grid that contains exactly two million rectangles, find the area of the grid with the nearest solution. | <p>By counting carefully it can be seen that a rectangular grid measuring $3$ by $2$ contains eighteen rectangles:</p>
<div class="center">
<img src="resources/images/0085.png?1678992052" class="dark_img" alt=""></div>
<p>Although there exists no rectangular grid that contains exactly two million rectangles, find the a... | https://projecteuler.net/problem=85 | 2772 |
86 | A spider, S, sits in one corner of a cuboid room, measuring $6$ by $5$ by $3$, and a fly, F, sits in the opposite corner. By travelling on the surfaces of the room the shortest "straight line" distance from S to F is $10$ and the path is shown on the diagram.
However, there are up to three "shortest" path candidates f... | <p>A spider, S, sits in one corner of a cuboid room, measuring $6$ by $5$ by $3$, and a fly, F, sits in the opposite corner. By travelling on the surfaces of the room the shortest "straight line" distance from S to F is $10$ and the path is shown on the diagram.</p>
<div class="center">
<img src="resources/images/0086.... | https://projecteuler.net/problem=86 | 1818 |
87 | The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is $28$. In fact, there are exactly four numbers below fifty that can be expressed in such a way:
$$\begin{align}
28 &= 2^2 + 2^3 + 2^4\\
33 &= 3^2 + 2^3 + 2^4\\
49 &= 5^2 + 2^3 + 2^4\\
47 &= 2^2 + 3^3 + 2^4
\end{align}$$
... | <p>The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is $28$. In fact, there are exactly four numbers below fifty that can be expressed in such a way:</p>
$$\begin{align}
28 &= 2^2 + 2^3 + 2^4\\
33 &= 3^2 + 2^3 + 2^4\\
49 &= 5^2 + 2^3 + 2^4\\
47 &= 2^2 + 3^... | https://projecteuler.net/problem=87 | 1097343 |
88 | A natural number, $N$, that can be written as the sum and product of a given set of at least two natural numbers, $\{a_1, a_2, \dots, a_k\}$ is called a product-sum number: $N = a_1 + a_2 + \cdots + a_k = a_1 \times a_2 \times \cdots \times a_k$.
For example, $6 = 1 + 2 + 3 = 1 \times 2 \times 3$.
For a given set of ... | <p>A natural number, $N$, that can be written as the sum and product of a given set of at least two natural numbers, $\{a_1, a_2, \dots, a_k\}$ is called a product-sum number: $N = a_1 + a_2 + \cdots + a_k = a_1 \times a_2 \times \cdots \times a_k$.</p>
<p>For example, $6 = 1 + 2 + 3 = 1 \times 2 \times 3$.</p>
<p>For ... | https://projecteuler.net/problem=88 | 7587457 |
89 | For a number written in Roman numerals to be considered valid there are basic rules which must be followed. Even though the rules allow some numbers to be expressed in more than one way there is always a "best" way of writing a particular number.
For example, it would appear that there are at least six ways of writing... | <p>For a number written in Roman numerals to be considered valid there are basic rules which must be followed. Even though the rules allow some numbers to be expressed in more than one way there is always a "best" way of writing a particular number.</p>
<p>For example, it would appear that there are at least six ways o... | https://projecteuler.net/problem=89 | 743 |
90 | Each of the six faces on a cube has a different digit ($0$ to $9$) written on it; the same is done to a second cube. By placing the two cubes side-by-side in different positions we can form a variety of $2$-digit numbers.
For example, the square number $64$ could be formed:
In fact, by carefully choosing the digits o... | <p>Each of the six faces on a cube has a different digit ($0$ to $9$) written on it; the same is done to a second cube. By placing the two cubes side-by-side in different positions we can form a variety of $2$-digit numbers.</p>
<p>For example, the square number $64$ could be formed:</p>
<div class="center">
<img src... | https://projecteuler.net/problem=90 | 1217 |
91 | The points $P(x_1, y_1)$ and $Q(x_2, y_2)$ are plotted at integer co-ordinates and are joined to the origin, $O(0,0)$, to form $\triangle OPQ$.
There are exactly fourteen triangles containing a right angle that can be formed when each co-ordinate lies between $0$ and $2$ inclusive; that is, $0 \le x_1, y_1, x_2, y_2 \... | <p>The points $P(x_1, y_1)$ and $Q(x_2, y_2)$ are plotted at integer co-ordinates and are joined to the origin, $O(0,0)$, to form $\triangle OPQ$.</p>
<div class="center">
<img src="resources/images/0091_1.png?1678992052" class="dark_img" alt=""><br></div>
<p>There are exactly fourteen triangles containing a right an... | https://projecteuler.net/problem=91 | 14234 |
92 | A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before.
For example,
$$\begin{align}
&44 \to 32 \to 13 \to 10 \to \mathbf 1 \to \mathbf 1\\
&85 \to \mathbf{89} \to 145 \to 42 \to 20 \to 4 \to 16 \to 37 \to 58 \to \mathbf{89}
\end{align}$... | <p>A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before.</p>
<p>For example,
$$\begin{align}
&44 \to 32 \to 13 \to 10 \to \mathbf 1 \to \mathbf 1\\
&85 \to \mathbf{89} \to 145 \to 42 \to 20 \to 4 \to 16 \to 37 \to 58 \to \mathbf... | https://projecteuler.net/problem=92 | 8581146 |
93 | By using each of the digits from the set, $\{1, 2, 3, 4\}$, exactly once, and making use of the four arithmetic operations ($+, -, \times, /$) and brackets/parentheses, it is possible to form different positive integer targets.
For example,
$$\begin{align}
8 &= (4 \times (1 + 3)) / 2\\
14 &= 4 \times (3 + 1 / 2)\\
19... | <p>By using each of the digits from the set, $\{1, 2, 3, 4\}$, exactly once, and making use of the four arithmetic operations ($+, -, \times, /$) and brackets/parentheses, it is possible to form different positive integer targets.</p>
<p>For example,</p>
$$\begin{align}
8 &= (4 \times (1 + 3)) / 2\\
14 &= 4 \ti... | https://projecteuler.net/problem=93 | 1258 |
94 | It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the almost equilateral triangle $5$-$5$-$6$ has an area of $12$ square units.
We shall define an almost equilateral triangle to be a triangle for which two sides are equal and the third differs by no more tha... | <p>It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the <dfn>almost equilateral triangle</dfn> $5$-$5$-$6$ has an area of $12$ square units.</p>
<p>We shall define an <dfn>almost equilateral triangle</dfn> to be a triangle for which two sides are equal and t... | https://projecteuler.net/problem=94 | 518408346 |
95 | The proper divisors of a number are all the divisors excluding the number itself. For example, the proper divisors of $28$ are $1$, $2$, $4$, $7$, and $14$. As the sum of these divisors is equal to $28$, we call it a perfect number.
Interestingly the sum of the proper divisors of $220$ is $284$ and the sum of the prop... | <p>The proper divisors of a number are all the divisors excluding the number itself. For example, the proper divisors of $28$ are $1$, $2$, $4$, $7$, and $14$. As the sum of these divisors is equal to $28$, we call it a perfect number.</p>
<p>Interestingly the sum of the proper divisors of $220$ is $284$ and the sum of... | https://projecteuler.net/problem=95 | 14316 |
96 | Su Doku (Japanese meaning number place) is the name given to a popular puzzle concept. Its origin is unclear, but credit must be attributed to Leonhard Euler who invented a similar, and much more difficult, puzzle idea called Latin Squares. The objective of Su Doku puzzles, however, is to replace the blanks (or zeros) ... | <p>Su Doku (Japanese meaning <i>number place</i>) is the name given to a popular puzzle concept. Its origin is unclear, but credit must be attributed to Leonhard Euler who invented a similar, and much more difficult, puzzle idea called Latin Squares. The objective of Su Doku puzzles, however, is to replace the blanks (... | https://projecteuler.net/problem=96 | 24702 |
97 | The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form $2^{6972593} - 1$; it contains exactly $2\,098\,960$ digits. Subsequently other Mersenne primes, of the form $2^p - 1$, have been found which contain more digits.
However, in 2004 there was found a mass... | <p>The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form $2^{6972593} - 1$; it contains exactly $2\,098\,960$ digits. Subsequently other Mersenne primes, of the form $2^p - 1$, have been found which contain more digits.</p>
<p>However, in 2004 there was fou... | https://projecteuler.net/problem=97 | 8739992577 |
98 | By replacing each of the letters in the word CARE with $1$, $2$, $9$, and $6$ respectively, we form a square number: $1296 = 36^2$. What is remarkable is that, by using the same digital substitutions, the anagram, RACE, also forms a square number: $9216 = 96^2$. We shall call CARE (and RACE) a square anagram word pair ... | <p>By replacing each of the letters in the word CARE with $1$, $2$, $9$, and $6$ respectively, we form a square number: $1296 = 36^2$. What is remarkable is that, by using the same digital substitutions, the anagram, RACE, also forms a square number: $9216 = 96^2$. We shall call CARE (and RACE) a square anagram word pa... | https://projecteuler.net/problem=98 | 18769 |
99 | Comparing two numbers written in index form like $2^{11}$ and $3^7$ is not difficult, as any calculator would confirm that $2^{11} = 2048 \lt 3^7 = 2187$.
However, confirming that $632382^{518061} \gt 519432^{525806}$ would be much more difficult, as both numbers contain over three million digits.
Using base_exp.txt ... | <p>Comparing two numbers written in index form like $2^{11}$ and $3^7$ is not difficult, as any calculator would confirm that $2^{11} = 2048 \lt 3^7 = 2187$.</p>
<p>However, confirming that $632382^{518061} \gt 519432^{525806}$ would be much more difficult, as both numbers contain over three million digits.</p>
<p>Usin... | https://projecteuler.net/problem=99 | 709 |
100 | If a box contains twenty-one coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken at random, it can be seen that the probability of taking two blue discs, $P(\text{BB}) = (15/21) \times (14/20) = 1/2$.
The next such arrangement, for which there is exactly $50\%$ chance of taking t... | <p>If a box contains twenty-one coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken at random, it can be seen that the probability of taking two blue discs, $P(\text{BB}) = (15/21) \times (14/20) = 1/2$.</p>
<p>The next such arrangement, for which there is exactly $50\%$ chance of... | https://projecteuler.net/problem=100 | 756872327473 |
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