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?