Suppose that , and are positive integers satisfying and . Show that there are integers and such that and .
I have some ideas for this. Trivially we can take and . I see nothing that says all numbers have to be different. (Although I suppose we could do this if is prime or a square leaving the below for when is composite and not a square)
Alternatively, write as a product of primes...
Then then we divide this up as we please such as...
since wont be divisible by and hence our condition holds.
How does this all look?