Задачи проекта Эйлера

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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.

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.
3. The prime factors of 13195 are 5, 7, 13 and 29.What is the largest prime factor of the number 600851475143 ?

4. A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 х 99.

   Find the largest palindrome made from the product of two 3-digit numbers.

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  (divisible witn no remainder) by all of the numbers from 1 to 20?

6. The sum of the squares of the first ten natural numbers is, 1² + 2² + ... + 10² =  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.

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?
8. Find the greatest product of five consecutive digits in the 1000-digit number.
   73167176531330624919225119674426574742355349194934
   96983520312774506326239578318016984801869478851843
   85861560789112949495459501737958331952853208805511
   12540698747158523863050715693290963295227443043557
   66896648950445244523161731856403098711121722383113
   62229893423380308135336276614282806444486645238749
   30358907296290491560440772390713810515859307960866
   70172427121883998797908792274921901699720888093776
   65727333001053367881220235421809751254540594752243
   52584907711670556013604839586446706324415722155397
   53697817977846174064955149290862569321978468622482
   83972241375657056057490261407972968652414535100474
   82166370484403199890008895243450658541227588666881
   16427171479924442928230863465674813919123162824586
   17866458359124566529476545682848912883142607690042
   24219022671055626321111109370544217506941658960408
   07198403850962455444362981230987879927244284909188
   84580156166097919133875499200524063689912560717606
   05886116467109405077541002256983155200055935729725
   71636269561882670428252483600823257530420752963450

9. A Pythagorean triplet is a set of three natural numbers, a <b < c, for which, a² + b² = c^2. For example, 3² + 4² = 9 + 16 = 25 = 5².

   There exists exactly one Pythagorean triplet for which a + b + c = 1000.
   Find the product abc.

10. The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.Find the sum of all the primes below two million.