Trouble setting up to prove unique factorization

[lants]

This lemma the book states, I can't make sense of it.

Lemma: If a,b\in Z and b > 0, there exist q,r \in Z such that a = qb + r with 0 \leq r < b.

Proof: Consider the set of all integers of the form a-xb with x \in Z. This set includes positive elements. Let r = a - qb be the least nonnegative element in this set. We claim that 0\leq r < b. If not, r = a - qb \geq b and so 0\leq a-(q+1)b<r, which contradicts the minimality of r.

Can someone help explain this to me? Why can't a = 3, and b = 8. Then no q,r could exist so that r is less than b, right?

 

[lants]

Is this considered a textbook style question? Sorry, how can I move it? I just saw number theory and posted

 

[Terandol]

What is wrong with a=3 and b=8? Here you simply take q=0 and r=3 (there is no condition that says either q or r have to be nonzero) to get 3=(0)(8) +3 and 0\leq 3&lt; 8 as required.

 

[lants]

Wow I'm quitting now

thanks