Prove that for any two positive integers n amd m,
Not sure how to approach this problem .
let d=gcd(n,m), then there exist positive integers k,s, such that n=kd,m=sd,(k,s)=1.
Obviously, is a common divisor of the two number.
And, it is easy to prove that and are relatively prime if (k,s)=1.
thus the result is proved!
Exactly,
Once you get to this part you can proceed as follows.
It suffices to show that there exist two numbers such that .
First, notice that
So suppose that then by definition there are positive numbers such that .
It is easy to verify that
and
satisfies the equation above, proving the result.
Thank you. However, is clear to me but my professor would like us to make a complete proof, so I would have to prove this @_@. How would one go about proving this as well?
I'm currently working on it as well though I'm stuck.
-Several hours later-
Still no closer to proving this.