Question

Сколькими нулями заканчивается произведение всех натуральных чисел от 10 до 50? ОБЪЯСНИТЕ!

Answer 1

10*11*...*20=6704425728000
21*22*...*30=109027350432000
31*32*...*40=3075990524006400
41*42*...*50=123840674496000
Итого 11 нулей

Answer 2

Calculating the Number of Zeroes in the Product of Natural Numbers from 10 to 50

To calculate the number of zeroes at the end of the product of all natural numbers from 10 to 50, we need to consider the prime factorization of each number and count the number of 2s and 5s, as the number of zeroes in the product will be determined by the minimum of the count of 2s and 5s in the prime factorization of each number.

The prime factorization of a number is the expression of that number as the product of prime numbers.

Prime Factorization of Numbers from 10 to 50

The prime factorization of each number from 10 to 50 is as follows: - 10 = 2 * 5 - 11 = 11 - 12 = 2^2 * 3 - 13 = 13 - 14 = 2 * 7 - 15 = 3 * 5 - 16 = 2^4 - 17 = 17 - 18 = 2 * 3^2 - 19 = 19 - 20 = 2^2 * 5 - 21 = 3 * 7 - 22 = 2 * 11 - 23 = 23 - 24 = 2^3 * 3 - 25 = 5^2 - 26 = 2 * 13 - 27 = 3^3 - 28 = 2^2 * 7 - 29 = 29 - 30 = 2 * 3 * 5 - 31 = 31 - 32 = 2^5 - 33 = 3 * 11 - 34 = 2 * 17 - 35 = 5 * 7 - 36 = 2^2 * 3^2 - 37 = 37 - 38 = 2 * 19 - 39 = 3 * 13 - 40 = 2^3 * 5 - 41 = 41 - 42 = 2 * 3 * 7 - 43 = 43 - 44 = 2^2 * 11 - 45 = 3^2 * 5 - 46 = 2 * 23 - 47 = 47 - 48 = 2^4 * 3 - 49 = 7^2 - 50 = 2 * 5^2

Counting the Number of Zeroes

The number of zeroes at the end of the product will be determined by the minimum of the count of 2s and 5s in the prime factorization of each number. Since the count of 5s will always be less than the count of 2s, we only need to count the number of 5s in the prime factorization of each number.

- For numbers less than 25, we count the number of 5s in their prime factorization. - For numbers between 25 and 50, we count the number of 5s and the additional 5s from the squares of 5.

Calculating the Number of Zeroes

The number of zeroes at the end of the product of all natural numbers from 10 to 50 can be calculated as follows:

10, 15, 20, 25, 30, 35, 40, 45, 50 each contribute one zero, and 25 contributes an additional zero due to the square of 5.

Therefore, the total number of zeroes at the end of the product of all natural numbers from 10 to 50 is 12.