Question

1. A unit fraction is a fraction of the form $\frac{1}{n}$, where $n$ is a positive integer. Note that the unit fraction $\frac{1}{11}$ can be written as the sum of two unit fractions in the following three ways: $\frac{1}{11} = \frac{1}{12} + \frac{1}{132} = \frac{1}{22} + \frac{1}{22} = \frac{1}{132} + \frac{1}{12}$. Are there any other ways of decomposing $\frac{1}{11}$ into the sum of two unit fractions? In how many ways can we write $\frac{1}{60}$ as the sum of two unit fractions? More generally, in how many ways can the unit fraction $\frac{1}{n}$ be written as the sum of two unit fractions? In other words, how many ordered pairs $(a, b)$ of positive integers $a, b$ are there for which $\frac{1}{n} = \frac{1}{a} + \frac{1}{b}$?

          1. A unit fraction is a fraction of the form $\frac{1}{n}$, where $n$ is a positive integer. Note that the unit fraction $\frac{1}{11}$ can be written as the sum of two unit fractions in the following three ways:
$\frac{1}{11} = \frac{1}{12} + \frac{1}{132} = \frac{1}{22} + \frac{1}{22} = \frac{1}{132} + \frac{1}{12}$.
Are there any other ways of decomposing $\frac{1}{11}$ into the sum of two unit fractions? In how many ways can we write $\frac{1}{60}$ as the sum of two unit fractions? More generally, in how many ways can the unit fraction $\frac{1}{n}$ be written as the sum of two unit fractions? In other words, how many ordered pairs $(a, b)$ of positive integers $a, b$ are there for which
$\frac{1}{n} = \frac{1}{a} + \frac{1}{b}$?

$1. A unit fraction is a fraction of the form (1)/(n), where n is a positive integer. Note that the unit fraction (1)/(11) can be written as the sum of two unit fractions in the following three ways: (1)/(11) = (1)/(12) + (1)/(132) = (1)/(22) + (1)/(22) = (1)/(132) + (1)/(12). Are there any other ways of decomposing (1)/(11) into the sum of two unit fractions? In how many ways can we write (1)/(60) as the sum of two unit fractions? More generally, in how many ways can the unit fraction (1)/(n) be written as the sum of two unit fractions? In other words, how many ordered pairs (a, b) of positive integers a, b are there for which (1)/(n) = (1)/(a) + (1)/(b)?$

Added by Danny A.

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