If a/b gives remainder r, and b/c also gives same remainder r => a/c would also give same remainder r for sure?

Question

Is this always true that:

If a/b gives remainder r, and b/c also gives same remainder r => a/c would also give same remainder r for sure?

Answer 1

You mean

$a:b=r$ and $b:c=r$,

or

$a=qb+r$ and $b=sc+r$, where $s,q$ are integers, $0\leq r<b$ and $0\leq r<c$. Then $a = q(sc+r)+r = qsc + r(q+1)$.

But 12:10 = 2, 10:8=2, but 12:8=4.