The problem states, if c^2 = ab and (a,b) = 1, prove that a and b are perfect squares.(adsbygoogle = window.adsbygoogle || []).push({});

( the notation (a,b) means the GCD of a and b)

So i have a lot of thoughts on this problem but i am getting stuck.

1) if c^2 = ab then c(c) = ab which says c divides ab.

2) the only real theorem available about the GCD of two numbers we have is that it can be written as a linear combination. so ax + by = 1 for some integers x and y.

3) however this problem is in the section about prime numbers and prime factorizations so i started thinking about perfect squares prime factorization

4 = 2^2

9 = 3^2

16 = 2^5 = 2^2(2^2)

25 = 5^2

36 = 2^2(3^2)

49 = 7^2

64 = 2^6 = 2^2(2^2)(2^2)

81 = 3^4 = 3^2(3^2)

and so on. so it seems this this should help me somehow. But i am not sure where to go with this or how i am going to use that fact that (a,b) = 1, so they are relatively prime.

Well wait if they are relatively prime, they couldn't be 2 perfect squares like 36 and 81 or else their GCD would not be 1 it would 3^2 or 9.

But a and b could be numbers like 25 and 16.

Anyone can help me finish please? A lot of these ideas came up as i type this.

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# Homework Help: Algebra proof involving perfect squares

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