Question

Пооомоогитеее алгебра 8 класс плиззз число а при делении на 3 дает остаток 2 а при делении на 5 ост.1. Какой остаток получится при делении этого числа на 15????

Answer 1

Problem Statement

You are given a number 'a' that leaves a remainder of 2 when divided by 3 and a remainder of 1 when divided by 5. You need to find the remainder when 'a' is divided by 15.

Solution

To find the remainder when 'a' is divided by 15, we can use the Chinese Remainder Theorem. According to the theorem, if we have a system of congruences of the form:

x ≡ a (mod m)
x ≡ b (mod n)

where 'm' and 'n' are coprime (i.e., they have no common factors other than 1), then there exists a unique solution for 'x' modulo 'm * n'.

In this case, we have the following congruences:

a ≡ 2 (mod 3)
a ≡ 1 (mod 5)

Since 3 and 5 are coprime, we can find the solution using the Chinese Remainder Theorem.

Chinese Remainder Theorem

To find the solution using the Chinese Remainder Theorem, we can follow these steps:

1. Find the modular inverses of 'm' and 'n' modulo 'n' and 'm', respectively. Let's call them 'r' and 's'. - In this case, 'm' is 3 and 'n' is 5. The modular inverse of 3 modulo 5 is 2, and the modular inverse of 5 modulo 3 is 2 as well.

2. Calculate 'x' using the following formula: ``` x = (a * n * s + b * m * r) % (m * n) ```

- In this case, 'a' is 2, 'b' is 1, 'm' is 3, 'n' is 5, 'r' is 2, and 's' is 2. Plugging these values into the formula, we get: ``` x = (2 * 5 * 2 + 1 * 3 * 2) % (3 * 5) = (20 + 6) % 15 = 26 % 15 = 11 ```

Therefore, the remainder when 'a' is divided by 15 is 11.

Conclusion

The remainder when the given number 'a' is divided by 15 is 11.