The goal of this exercise is to prove the following theorem in several steps. Theorem: Let...

The goal of this exercise is to prove the following theorem in several steps.
Theorem: Let ? and ? be natural numbers. Then, there exist unique
integers ? and ? such that ? = ?? + ? and 0 ≤ ? < ?.
Recall: that ? is called the quotient and ? the remainder of the division
of ? by ?.
(a) Let ?, ? ∈ Z with 0 ≤ ? < ?. Prove that ? divides ? if and only if ? = 0.
(b) Use part (a) to prove the uniqueness part of the theorem. That is, show thatiftherearetwopairs? ,? ∈Zand? ,? ∈Zsatisfying?=? ? +
11221
?,0≤? <?,and?=? ?+?,0≤? <?,then? =? and? =?. 112221212
(c) Prove that there exist such ? and ? when ? divides ?.
(d) Prove that there exist such ? and ? when ? does not divide ? by applying the Well-Ordering Principle to the set
? = {? ∈ N: ? = ? − ?? ??? ???? ? ∈ Z}.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

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