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
That is as much as I can get. How can I show that ?
Any hints would be appreciated.