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?