(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If a and b are positive integers, then ab=gcd(a,b)*lcm(a,b).

2. Relevant equations

I am allowed to use the following propositions which have already been proved:

(1) If d is a common divisor of a and b, then ab/d is a common multiple of a and b.

(2) If m is a common multiple of a and b and m divides ab, then ab/m is a common divisor of a and b.

A hint given:

set d=gcd(a,b) and m=lcm(a,b). Use (1) to show that ab/d>=m. Use (2) to show that ab/m<=d.

3. The attempt at a solution

1. Let a and b be positive integers. Suppose d=gcd(a,b) and m=lcm(a,b).

2. By (1) ab/d is a common multiple of a and b so ab/d=aL and ab/d=bK

3. Multiply by m gives mab/d=aLm and mab/d=bKm

4. ab/d>=m

I am missing a step between 2 and 3. Any suggestions?

Thanks,

Scott

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# Homework Help: Solve ab=gcd(a,b)*lcm(a,b)

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