Projet Euler

Voici quelques problèmes tirés du Projet Euler: https://projecteuler.net. Calculez les réponses aux questions et vérifiez sur le site que vous avez trouvé la solution. Vous pouvez rajouter d'autres problèmes après ceux proposés. A vous d'écrire vos propres programmes ou de chercher dans Sage les fonctionalités dont vous avez besoin.

Problème 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.

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### Problème 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, ...

By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.

Problème 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, ...

By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.

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### Problème 3

The prime factors of 13195 are 5, 7, 13 and 29.

What is the largest prime factor of the number 600851475143 ?

Problème 3

The prime factors of 13195 are 5, 7, 13 and 29.

What is the largest prime factor of the number 600851475143 ?


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︠e0ff342d-7de9-4d9a-9ebc-825d1b61683ei︠
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### Problème 4

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.

Problème 4

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.

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︠2fcaf775-e9e5-437b-8e45-9fa33cdfe916i︠
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### Problème 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 divisible by all of the numbers from 1 to 20?

Problème 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 divisible by all of the numbers from 1 to 20?


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︠5da920ca-7125-4284-9e90-82cecc3e16dd︠

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### Problème 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,
$(1 + 2 + ... + 10)^2 = 552 = 3025$

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 hundred natural numbers and the square of the sum.

Problème 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, $(1 + 2 + ... + 10)^2 = 552 = 3025$

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 hundred natural numbers and the square of the sum.

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### Problème 7

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.

What is the 10 001st prime number?

Problème 7

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.

What is the 10 001st prime number?

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︠8a9ee123-76c3-4ea9-86ef-069225e80385︠

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### Problème 8

The four adjacent digits in this 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832. Find the thirteen adjacent digits in this 1000-digit number that have the greatest product. What is the value of this product?

Problème 8

The four adjacent digits in this 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832. Find the thirteen adjacent digits in this 1000-digit number that have the greatest product. What is the value of this product?

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### Problème 9

A Pythagorean triplet is a set of three natural numbers, $a < b < c$, 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.

Problème 9

A Pythagorean triplet is a set of three natural numbers, $a < b < c$ , 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.

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### Problème 10

The sum of the primes below 10 is $2 + 3 + 5 + 7 = 17$.

Find the sum of all the primes below two million.

Problème 10

The sum of the primes below 10 is $2 + 3 + 5 + 7 = 17$ .

Find the sum of all the primes below two million.

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