ATYPIC

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which, a2 + b2 = c2 For example, 32 + 42 = 9 + 16 = 25 = 52. There exists exactly one Pythagorean triplet for which a + b + c = 1000. Find the product ABC. 더보기 더보기 Let's start with some math. Since $a + b + c = 1000$, $c$ can be substituted by $1000 - a - b$. \[a^2 + b^2 = c^2 = (1000 - a - b)^2\] Then: \[\begin{align}a^2 + ..

The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832. 73167176531330624919225119674426574742355349194934 96983520312774506326239578318016984801869478851843 85861560789112949495459501737958331952853208805511 12540698747158523863050715693290963295227443043557 66896648950445244523161731856403098711121722383113 622298934233803081353362766142828064..

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 $3025 - 385 = 2640$. Find the difference between the sum of the squares of the first one hund..

The prime factors of 13195 are 5, 7, 13, and 29. What is the largest prime factor of the number 600851475143? 더보기 Prime number generator. # Sieve of Eratosthenes # Code by David Eppstein, UC Irvine, 28 Feb 2002 # http://code.activestate.com/recipes/117119/ def gen_primes(): """ Generate an infinite sequence of prime numbers. """ # Maps composites to primes witnessing their compositeness. # This ..

Comparing two numbers written in index form like 211 and 37 is not difficult, as any calculator would confirm that 211 = 2048 519432525806 would be much more difficult, as both numbers contain over three million digits. Using base_exp.txt, a 22K text file containing one thousand lines with a base/exponent pair on each line, determine which lin..

Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of numbers less than 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, φ(9)=6. n Relatively Prime φ(n) n/φ(n) 2 1 1 2 3 1, 2 2 1.5 4 1, 3 2 2 5 1, 2, 3, 4 4 1.25 6 1, 5 2 3 7 1, 2, 3, 4, 5, 6 6 1.1666… 8 1, 3, 5, 7 4 2 9 1..

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 divisible by all of the numbers from 1 to 20? 더보기 This is a problem about getting an LCM of all the numbers from 1 to 20. To do so, I have to implement a prime factorization function. First of all, find the prime numbers. Let's use the tra..

A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99. Find the largest palindrome made from the product of two 3-digit numbers. 더보기 The problem requires the largest, so I'll go backward. for i in range(999, 99, -1): p = int(str(i) + str(i)[::-1]) This gives all the palindromes from 999999 to 100001 (If I have to find..

In the 5 by 5 matrix above, 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 & \color{red}{746} & \color{red}{422} & 111\\ 537 & 699 & 497 & \color{red}{121} & 956\\ ..

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, … By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms. 더보기 Start with the Fibonacci sequence generator. a, b = 1, 2 while True: a, b = b, a + b ..