Characteristic of a product ring?

Question

Let A and B be commutative rings with unity where char(A)=n and Char(B)=m s.t. n,m ∈ ℤ (and n,m ≠ 0). Prove or give counter example: if k ∈ ℤ⁺ and n,m both divide k, then Char (A x B)| k.

Here was my thinking but I don't know if it is true: Char(A x B) = gcd(n,m) . I know that if n|k and m|k, we want to show that gcd(n,m)|K. However, I don't know if my initial assumption that Char(A x B) = gcd(n,m) is actually correct.

I know that a characteristic of a ring with unity is the smallest such n st 1*n≡ 0 where 1 is the multiplicative identity. I tried to use an easy example:

Char(ℤ₁₂) = 12, Char(ℤ₆)=6, Char(ℤ₁₂ x ℤ₆) = 3 or does it equal 6???

Answer 1

As you were told in the comments, $\mathrm{char}(A\times B)=\mathrm{lcm}(m,n)$. (To see why take a look at Group Order and Least Common Multiple.) So the property you are looking for holds true.