My understanding of the division theorem: Let $a,b$ be integers, $b>0$. Then there are unique integers q,r such that $a=qb+r$, and $0\leq r<b$
Is this supposed to hold for any arbitrarily chosen integer values for $a$ and $b$?
Example: $a= 1, b =5$. Then $1=5q +r$. If $q$ is supposed to be an integer, and $r$ cannot be negative and must $<b$, then unless I'm missing something very basic this statement cannot hold. So where has my understanding of the division theorem gone wrong?
