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Positive integer solutions

How many triple of positive integer \((a,b,c)\) that satisfy equation: \[\frac{c}{17}=\frac{8}{a^2}+\frac{45}{b^2}\]

Note by Idham Muqoddas
3 years, 2 months ago

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Rearranging the equation we get: \[c=\frac{17\cdot 2^3}{a^2}+\frac{17 \cdot 3^2 \cdot 5}{b^2}\] Because the LHS is an integer, the RHS must also be an integer. First, let's address the case in which both fractions in the RHS are integers. Then \(a=1, 2\) and \(b=1, 3\), giving us \(4\) solutions so far.

Now note that when \(a > 11\) and \(b > 27\), the RHS is less than one, and thus cannot be a positive integer. You could easily exhaust all 297 possibilities, although I'm sure there is a better way. Bob Krueger · 3 years, 2 months ago

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