Question

Consider the homomorphism ∅: Z/mnZ → Z/mZ x Z/nZ, used in the Chinese Remainder Theorem.

Prove that if GCD(m, n) = 1, then LCM(m, n) = mn. Hint: if ∅ (x + mnZ) = (0+mZ, 0+nZ) then use injectivity of f to conclude that x is a multiple of mn.

Answer 1

The statement is proven: if GCD(m, n) = 1, then LCM(m, n) = mn.

Let's prove this statement step by step.

Given the homomorphism Ø: Z/mnZZ/mZxZ/nZ used in the Chinese Remainder Theorem.

Assume Ø(x + mnZ) = (0+mZ, 0+ nZ).

We know that Ø is a homomorphism. This implies that if (x + mnZ) = (0+ mZ, 0+ nZ), then x + mnZ = 0 + mnZ (using the properties of the homomorphism). This means 2 is a multiple of mn.

Now, let's consider the property that GCD(m, n) = 1, which implies that m and n are coprime.

Recall that for any two integers m and in that are coprime, LCM(m, n) = mn.

If GCD(m, n) = 1, then LCM(m, n) = mn.

Therefore, the statement is proven: if GCD(m, n) = 1, then LCM(m, n) = mn.