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 â¥ 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

Question

Mathematics

Milton Wood

21 September, 02:22

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 â¥ 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

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Terry Harrington · Answer 1

r1=r2

Step-by-step 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.