Let a and b be nonzero integers. Prove that there exist a unique q, r also integers such that a=bq+r and 0<=r<|b|

Question

Let a and b be nonzero integers. Prove that there exist a unique q, r also integers such that a=bq+r and 0 <= r<|b|

The number theory book ask me to prove this but I realize that r can be <0 for example take a= 10 and b=-3 then 10/-3=-3.3333333 therefore q must =-4 because q <= a/b as a result q*b =12 and then r must be negative 2. Where did I go wrong?

I apologize for the format I tried using \le but it did not work and I wanted this to be understandable.

Thanks

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Answer 1

The error is: Nobody said $q\le\frac ab$.

Answer 2

You cannot take $r=\frac{10}{-3}$. It's written in the statement that $r$ must be an integer.