Question

Assume a and b are both integers and a > 0. Define a remainder after the division of b by a to be a value r such that r ⥠0, r < a, and there exists an integer q for which b = aq + r. a) Prove uniqueness. That is, if r1 and r2 are both remainders after the division of b by a, then r1 = r2. Y

Answer 1

r1=r2

Explanation:

Initially,let´s assume there are two remainders r1, and r2. This must fit the following equations:

b=a.q + r1 (equation 1)

b=a.q + r2 (equation 2)

Because we have the same variable b in both equations, we can make equal both expressions, and solving:

a.q + r1 = a.q +r2

- r2= -r2

a.q + r1 - r2 = a.q + 0

a. q + r1 - r2 = a.q

-a.q = - a.q

0 + r1 - r2 = 0

r1 - r2 = 0

+r2 = r2

r1 + 0 = r2

r1 = r2 And this is what we want to proof.