Why for 7 | 5m+2n you can just make 7 | 5n+2n - 7n

Question

Why for 7 | 5m+2n you can just make 7 | 5n+2n - 7n Why can you just add te 7n, and is it only true for whole numbers = n. Thank you for your answer

Answer 1

It's the same thing used in long division. You can use known multiples of your divisor, and not change the remainder on division by that divisor, at the end of the long division sequence.

Answer 2

Since $7\mid 5m+2n$, we can write $5m+2n=7k$ for some $k$. Then $$ 5m+2n-7n=7k-7n=7(k-n) $$ which shows that $7\mid 5m+2n-7n$.