   Chapter 2.4, Problem 24E

Chapter
Section
Textbook Problem

Let ( a ,   b ) = 1 . Prove that ( a ,   b n ) = 1 for all positive integers n .

To determine

To prove: If (a,b)=1 then, (a,bn)=1, for all positive integers n.

Explanation

Given information:

(a,b)=1

Formula used:

If p is a prime and p|an, then p|a.

Proof:

Let (a,b)=1.

Suppose, (a,bn)=d1.

Since, d1, then, there exists a prime p such that p|d.

As, d=(a,bn)d|a and d|bn.

Since, p|d and d|ap|a.

Similarly, since p|d and d|bnp|bn

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