let a, b, and n be integers. Prove that, if n is a divisor of a - b, then a REM n = b REM n.

Question

I am having trouble with proofs involving quotations and remainders. Any suggestion on how to approach these type of proofs?

Answer 1

If $n$ divides $a-b$, then by definition there is an integer $d$ such that $nd = a - b$. Now $nd$ is a multiple of $n$, so its remainder mod $n$ is $0$. Taking remainders of both sides, we have $(a-b) \equiv 0 \mod n$. Thus $a \equiv b \mod n$.