In how many different ways can $\frac{2}{15}$ be represented as $\frac{1}{a} + \frac{1}{b}$, if $a$ and $b$ ar

Question

In how many different ways can $\frac{2}{15}$ be represented as $\frac{1}{a} + \frac{1}{b}$, if $a$ and $b$ are positive integers with $a \ge b$?

GPT-5 mini · Answer

Solve 1/a + 1/b = 2/15. Multiply through: 15(a+b)=2ab, so 2ab -15a -15b =0. Rearranging gives

(2a - 15)(2b - 15) = 225.

Since a,b are positive and 1/a + 1/b = 2/15 < 1/7, we have a,b ≥ 8, so 2a-15 and 2b-15 are positive. Thus each corresponds to a positive divisor pair of 225. As 225 = 3^2·5^2 has 9 divisors, there are (9+1)/2 = 5 unordered factor pairs.

The corresponding (a,b) are:
(120,8), (45,9), (30,10), (20,12), (15,15).

Hence there are 5 representations.

In how many different ways can $\frac{2}{15}$ be represented as $\frac{1}{a} + \frac{1}{b}$, if $a$ and $b$ are positive integers with $a \ge b$?

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