Discrete Math Modulus Beginner Direct Proof

Question

I am having trouble with a direct proof in my discrete mathematics class. It is about modularity.

The statement I am supposed to prove is, "If a is congruent to b mod 12, then a is congruent to b mod 6."

I understand that there are 3 things that are immediately known when doing a direct proof pertaining to modulus.

1. a = 12k + b

2. a = 12q + r b = 12n + r

3. a - b = 12m

With that being said, I have been staring at this problem for a couple of hours, and I know the solution is probably staring right at me in the face, but I have been unable to find it. Any help or hints would be greatly appreciated. Thanks.

Answer 1

"$a$ is congruent to $b$ modulo 12" means that $r=\frac1{12}(a-b)$ is an integer.

"$a$ is congruent to $b$ modulo 6" means that $s=\frac16(a-b)$ is an integer.

So you need to prove that if $r$ is an integer, then $s$ is an integer.

What is the relationship between $r$ and $s$?