
LCM and GCD problem
I had to figure out the following lcms and gcds and compare the two:
I got that
LCM(8,12)=24 and GCD(8,12)=4
LCM(20,30)=60 and GCD(20,30)=10
LCM(51,68)=204 and GCD(51,68)=17
LCM(23,18)=414 and GCD(23,18)=1
The pattern I found was that if you take the first number enclosed in the parentheses, divide it by the GCD, and multiply it by the second number in the parentheses, you get the LCM. For example, 8/4=2, which when multiplied by 12, gives you the LCM of 24.
My question is, how do I show that this relationship holds true for all m and n (where m and n are the numbers between the parentheses)?

Well, I may not understand you difficulty correctly.
However, this is a well known theorem: [LCM(x,y)][GCF(x,y)]=xy.
This proved by the prime factorization theorem.