Question

Сколько раз произведение всех чисел от 1 до 50 (50!) можно без остатка разделить на число 7?

Answer 1

1) 7 / 7 = 1;

2) 14 / 7 = 2;

3) 21 / 7 = 3;

4) 28 / 7 = 4;

5) 35 / 7 = 5;

6) 42 / 7 = 6;

7) 49 / 7 = 7;

7 раз мы можем это сделать

Answer 2

Calculating 50! Divisible by 7

To determine how many times the product of all numbers from 1 to 50 (50!) can be divided by 7 without a remainder, we can use the concept of prime factorization.

The prime factorization of 50! can help us determine the number of times it can be divided by 7 without a remainder.

The prime factorization of 50! is calculated as follows:

1. Prime Factorization of 50!: - The prime factorization of 50! can be calculated by finding the prime factors of each number from 1 to 50 and then multiplying them together.

2. Finding the Number of Times 50! is Divisible by 7: - We need to find the power of 7 in the prime factorization of 50! to determine how many times 50! can be divided by 7 without a remainder.

3. Calculation: - The number of times 50! can be divided by 7 without a remainder can be determined by finding the sum of the quotients obtained when dividing 50 by 7, 50 by 7^2, 50 by 7^3, and so on, until the quotient becomes 0.

Prime Factorization of 50!

The prime factorization of 50! can be calculated using the prime factors of each number from 1 to 50.

Prime Factorization of 50!:

50! = 2^47 * 3^22 * 5^12 * 7^8 * 11^4 * 13^3 * 17^2 * 19^2 * 23 * 29 * 31 * 37 * 41 * 43 * 47.

Number of Times 50! is Divisible by 7

To find the number of times 50! can be divided by 7 without a remainder, we calculate the sum of the quotients obtained when dividing 50 by 7, 50 by 7^2, 50 by 7^3, and so on, until the quotient becomes 0.

Calculation:

50 ÷ 7 = 7 (quotient) 50 ÷ 7^2 = 1 (quotient) 50 ÷ 7^3 = 0 (quotient)

The sum of the quotients is 7 + 1 = 8.

Conclusion: The product of all numbers from 1 to 50 (50!) can be divided by 7 without a remainder 8 times.

Therefore, 50! is divisible by 7 8 times.