   Chapter 8.4, Problem 30ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
1 views

# Finish the proof of Theorem 8.4.5 by proving that if a,b, and c are as in the proof, then c | b .

To determine

To prove:

a,bandc are integers then c|d.

Explanation

Given information:

a,bandc are integers such that c=as+tb for some integers s and t.

Here, c is the least element in S={x|xisapositiveintegerandx=as+bt}.

Formula used:

Division algorithm:

Let a be an integer and d be a positive integer. Then, there are unique integers qandr with 0r<d such that a=dq+r, here, q is the quotient and r is the remainder.

a divides b if there exists an integer c such that b=ac. The notation is a|b

Proof:

By the division algorithm, there exist integers qandr with 0r<c such that b=cq+r.

Subtract cq from both side of equation b=cq+r

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