Prove a and b are Perfect Squares

Date: 12/28/2001 at 01:09:13
From: salman
Subject: Number theory

Hello, Dr. Maths,

Please help me in solving the folowing question:

Let a and b be positive integers such that (a,b) = 1 and ab is a 
perfect square. Prove that a and b are perfect squares.

Thanks.

Date: 01/02/2002 at 15:47:02
From: Doctor Rob
Subject: Re: Number theory

Thanks for writing to Ask Dr. Math, Salman.

Write a = A^2*x and b = B^2*y, where x and y are square-free.  Then
a*b = A^2*B^2*(x*y) is a perfect square, so x*y must be a perfect
square.  But (a,b) = 1, so (x,y) = 1.  If p is any prime divisor of
x*y, then p^2 | x*y, because x*y is a perfect square.  

Now since x and y are squarefree, it is not possible to have p^2 | x 
or p^2 | y. That means that both p | x and p | y. But that implies 
that p | (x,y) = 1, a contradiction. The conclusion is that there is 
no prime divisor of x*y, so x*y = 1, x = y = 1, and a = A^2 and 
b = B^2 are both perfect squares.

Feel free to write again if I can help further.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/