Prove a and b are Perfect SquaresDate: 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/ |
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