<s>[INST] How do I prove that any unit fraction can be represented as the sum of two other distinct unit fractions? A number of the form $\frac{1}{n}$, where $n$ is an integer greater than $1$, is called a unit fraction.
Noting that
$\frac{1}{2} = \frac{1}{3} + \frac{1}{6}$
and
$\frac{1}{3} = \frac{1}{4} + \frac{1}{12}$,
find a general result of the form
$\frac{1}{n} = \frac{1}{a} + \frac{1}{b}$
and hence prove that any unit fraction can be expressed as the sum of two other distinct unit
fractions.
 [/INST] $\frac{1}{n}$ - $\frac{1}{n + k}$ = $\frac{k}{n(n+k)}$
$\frac{k}{n(n+k)}$ = $\frac{1}{\frac{n(n+k)}{k}}$
So, as long as $\frac{n(n+k)}{k}$ is a whole number, we can say that $\frac{1}{\frac{n(n+k)}{k}}$ is a unit fraction and since
$\frac{1}{n}$ = $\frac{k}{n(n+k)}$ +   $\frac{1}{n + k}$
$\frac{1}{n}$  can be expressed as a sum of two unit fractions as long as k has some value such that $\frac{n(n+k)}{k}$ is a whole number.
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