Good day to all. I have been asked to prove the following proposition:
Let m and n be integers. Use the contradiction method to prove that the quotient and the remainder of the division of m by n are unique.
The following is a rough proof I have come up with. I was wondering if someone could offer their opinion on my proof, if I have made glaring mistakes and such. Any input would be greatly appreciated. I have omitted the theorems used, to avoid making the post heavier than it already is.
We assume that the quotient and remainder are not unique. Then
m = q1n + r1 and m = q2n + r2
q1n + r1=q2n + r2
n(q1-q2) = -(r1-r2)
This implies the equality 0 = 0 since zero is the only number equal to its negative.
Therefore n(q1-q2) = 0 => n = 0 or (q1-q2) = 0
if n = 0 => r1 = r2 and we have q1 = ((n-r1)/n) and q2 = ((n-r1)/n) i.e. q1 = q2
if (q1-q2) = 0 => q1 = q2 and -(r1-r2) = 0 => r1 = r2