# Tag Archives: division algorithm

## Exam Review 2 UPDATE – review of division and remainders

Hi everyone,

First, the answer key for the Exam 2 review is now complete (it follows the questions in the Review document).  Second, you will notice that there are a few places (especially in problems 8 and 12) that I use basic facts about remainders when one number is divided by another.  These are facts that you understand intuitively from working with dividing numbers, but it might help to have them stated explicitly.  This is the basic fact about dividing and remainders:

Fact (The Division Algorithm).  Given two integers a and b with $b>0$, there exist unique integers q and r for which $a=qb+r$ and $0\leq r .

What does this mean?  It expresses the fact that when we divide one integer a by another integer b, and it will go in some number of times q with a remainder of r.  The remainder r must be less than the number b that we are dividing by.  If the remainder is zero, then b divides a (because $r=0$ means $a=qb$), and if the remainder is not zero, then b does not divide a.