If $\displaystyle a$ and $\displaystyle b$ are natural numbers, $\displaystyle (a,b) = 1$, and $\displaystyle ab$ is square, show that $\displaystyle a$ and $\displaystyle b$ are squares.
If $\displaystyle a$ and $\displaystyle b$ are natural numbers, $\displaystyle (a,b) = 1$, and $\displaystyle ab$ is square, show that $\displaystyle a$ and $\displaystyle b$ are squares.
ab has a unique prime number decomposition. Since ab is even, every prime number is to an even power. Now look at the prime number decompositions of a and b. Since (a,b)= 1, they have NO prime numbers in common.