It'll be easier if you think in terms of prime factorizations, i.e. if
and , with all p's and q's prime, and all s's, r's positive integers,
how would you express the relationships and ?
In this thread:
http://www.mathhelpforum.com/math-he...an-triple.html
I was trying to prove that if d^2 divides z^2, then d divides z.
It seems like it should be an easy proof, but the proof I came up with is long and unclear (maybe even wrong). Can anyone come up with a better proof?
Here it is:
By definition, means there is a positive integer n such that . Suppose n is not the square of an integer. Let , where m>1 has no square divisors. Let p be a prime divisor of m. We have , and since the powers of p in , , and are even, the power of p in m must be even. But then m has a square factor. By contradiction, therefore, n must be the square of an integer. Let . Then and since z and d are positive and you can choose a to be positive, z=ad, which is the definition of .
So:
means that if then
and:
means that if then
which are obviously equivalent statements.
I was looking for something more like the proof of for any integers m,n. From the definition of divisibility, there are integers r,s such that b=ra and c=sa. Then mb+nc = (mr+ns)a and so a divides mb+nc.
Maybe what I'm looking for doesn't exist...