Originally Posted by

**ScottO** Hi Folks,

I'm having difficulty making progress on the following proof, and I'm hoping for some hints to help me along.

Let $\displaystyle b$ denote a fixed positive integer. Prove the following statement by induction: For every integer $\displaystyle n \geq 0$, there exist non-negative integers $\displaystyle q$ and $\displaystyle r$ such that

$\displaystyle n = qb + r, 0 \leq r < b$.

So I can state the assertion and show the initial case is true:

$\displaystyle A(n): n = qb + r, 0 \leq r < b$

$\displaystyle A(0): 0 = 0 \cdot b + 0, n = q = r = 0$

I've tried picking a value for b and creating a table of related values for q and r as n counts up from 0. But I can't figure out how to express the patterns I see to form a usable general case and inductive step.

Again, just looking for hints right now to help get me out of the rut I'm in.

Thanks,

Scott