$a$ and $b>5$ are natural numbers, $a:b$ gives a remainder $(b-5)$, what will be the remainder when $(2a+7):b$?

Question

Question: $a$ and $b>5$ are natural numbers, $a:b$ gives a remainder $(b-5)$, what will be the remainder when $(2a+7):b$?

How to solve this?

Answer 1

Since $$a = kb+b-5,$$ for some integer $k$, we have $b\geq 5$ and $$2a+7 = 2kb+2b-10+7 = b(2k+1)+b-3$$

So a remainder is $b-3$.