Theorem C $($Division Algorithm for Integers$).$ Let a and $b$ be integers, with $b$ nonzero.

Then there exist integers $q$ and $r$ such that

$a=bq+r,$ and

Moreover, if $q,q^{\text{'}}$ and $r,r^{\text{'}}$ are any integers that satisfy

$a=bq+r$

$=b{q}^{\text{'}}+r^{\text{'}}$

with $0r<mi>|</mi><mi>b</mi><mi>|</mi>$ and then $q=q^{\text{'}}$ and $r=r^{\text{'}}.$