   Chapter 2.4, Problem 29E

Chapter
Section
Textbook Problem

Let a and b be positive integers. Prove that if d = ( a ,   b ) , a = a 0 d , and b = b 0 d , then the least common multiple of a and b is a 0 b 0 d .

To determine

To prove: If d=(a,b),a=a0d and b=b0d, then the least common multiple of a and b is a0b0d.

Explanation

Given information:

a and b are positive integers. d=(a,b),a=a0d and b=b0d.

Formula used:

1) Least Common Multiple:

A least common multiple of two non-zero integers a and b is an integer m that satisfies all the following conditions:

1. m is a positive integer.

2. a|m and b|m.

3. a|c and b|c imply m|c.

2) Let a and b be positive integers. If d=(a,b) and m is the least common multiple of a and b, then, dm=ab.

Proof:

Let a and b be positive integers with d=(a,b),a=a0d and b=b0d.

Since, d=(a,b)d|a and d|b

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