# 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
4 years, 10 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.

- 4 years, 10 months ago