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# A B C and fractions

How many possible value of $$c <100$$, so that

$\dfrac{1}{c} = \dfrac{1}{a} + \dfrac{1}{b}$

only have $$9$$ solutions for $$(a,b)$$.

Detail : $$( a,b,c ) > 0$$

Note by Fidel Simanjuntak
7 months, 4 weeks ago

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Here is another hint:

The number of positive integer solutions for $$\frac {1}{a}+\frac {1}{b}=\frac {1}{n}$$ is the number of factors of $$n^2$$

- 7 months, 3 weeks ago

Hint: Show that a divides b (or b divides a) first.

And since there's 9 (odd number) solutions, then there's a solution for which a=b, so c is even.

- 7 months, 4 weeks ago