Thread: Help with a problem involving the GCD, LCM.

Let a and b be positive integers, and let m be an integer such that Without using the prime factorization theorem, prove that by verifying that m satisfies the necessary properties of

Note that denotes the GCD of a and b and denotes the LCM of a and b.

Here is the definition for LCM that I have (the two properties that I need to show.)

A positive integer m is called the least common multiple of the nonzero integers a and b if

(i) m is a multiple of both a and b, and

(ii) any multiple of both a and b is also a multiple of m.

Property (i) was not very difficult to show, but I am struggling with the second one. Here is what I have so far.

Let and , where

Then and where

and

and

and

and

and

That is as much as I can get. How can I show that ?

Any hints would be appreciated.

Edit: See here

Basic Number Theory: LCM/GCD Proof

(Make sure to read posts #1 through #4 to get a full treatment of part (2).)

Although the proof for part (1) is lengthy for my taste, since



And since (a,b) | b, we know is an integer, hence a | m, and by symmetry b | m also.

Edit 2: I think aman_cc's proof for part (2) below is nicer than the one on the other forum.

Hint use the fact that

xa + yb = (a,b) for some integers x,y
xaq + ybq = (a,b)q
divide the whole eq by ab
xq/b + yq/a = q/m
it should be easy from here