Assume we have integers a, b, and c such that c =ab and gcd(a,b)=1. Show that c is a perfect square if and only if a and b are perfect squares.

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- Nov 7th 2007, 11:13 AM #1

- Nov 7th 2007, 11:44 AM #2

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- Nov 7th 2007, 04:54 PM #4

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Let $\displaystyle a=p_1^{a_1}...p_n^{a_n}$ and $\displaystyle b=q_1^{b_1}...q_m^{a_m}$ so if $\displaystyle ab = p_1^{a_1}...q_m^{b_m}$ is a square it means all exponents $\displaystyle a_1,a_2,...,b_m$ are even so $\displaystyle a$ and $\displaystyle b$ have even exponents and so are squares.

- Nov 7th 2007, 05:04 PM #5

- Nov 7th 2007, 05:21 PM #6

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- Nov 7th 2007, 05:23 PM #7

- Nov 7th 2007, 05:24 PM #8

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- Nov 7th 2007, 05:42 PM #9