# Tag Archives: remainder

## 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). Ā Second, you will notice that there are a few places (especially in problems 8 and 12) that I use basic facts aboutĀ remainders when one number is divided by another. Ā These are facts that you understand intuitively from working with dividing numbers, but it might help to have them stated explicitly. Ā This is the basic fact about dividing and remainders:

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

What does this mean? Ā It expresses the fact that when we divide one integerĀ a by another integerĀ b, and it will go in some number of timesĀ qĀ with a remainder ofĀ r. Ā The remainder rĀ must be less than the numberĀ b that we are dividing by. Ā If the remainder is zero, then b dividesĀ a (becauseĀ $r=0$ means $a=qb$), and if the remainder is not zero, thenĀ b does not divideĀ a.