Question

Find the smallest positive integer solution to the following system of congruence:

[ x ≡ 0 (mod 11) ]
[ x ≡ 34 (mod 47) ]

Answer 1

The smallest positive integer solution to the system of congruences x ≡ 0 (mod 11) and x ≡ 34 (mod 47) is found using the Chinese Remainder Theorem, with the result being x = 330.

Step-by-step explanation:

The student is asking for the smallest positive integer solution to a system of congruences: x ≡ 0 (mod 11) and x ≡ 34 (mod 47). To solve this, we need to use the Chinese Remainder Theorem, which is a method for solving systems of simultaneous congruences with different moduli that are coprime.

First, we acknowledge that x must be a multiple of 11. We then find a multiple of 11 that, when divided by 47, leaves a remainder of 34. By trial and error or by using an algorithm, we can find such a multiple. For instance, 11 × 30 = 330, and 330 ≡ 34 (mod 47),

Therefore, the smallest positive integer that satisfies both congruences is x = 330.