Question

The solution of the system of congruence x = 3(mod 5), x = 5(mod 7) is

x =29 mod 35
x =27 mod 35
x =23 mod 35
x =33 mod 35

Answer 1

To solve this system of congruences, we can use the Chinese Remainder Theorem.

From the first congruence, we know that x is of the form x = 5k + 3, where k is an integer.

Substituting this into the second congruence, we get:

5k + 3 ≡ 5 (mod 7)

5k ≡ 2 (mod 7)

Multiplying both sides by the inverse of 5 modulo 7, which is 3, we get:

k ≡ 6 (mod 7)

So, k is of the form k = 7m + 6, where m is an integer.

Substituting this back into x = 5k + 3, we get:

x ≡ 5(7m + 6) + 3 (mod 35)

x ≡ 35m + 33 (mod 35)

Therefore, the solution to the system of congruences is x ≡ 33 (mod 35).

So, the answer is x = 33 mod 35.