[ $\quad$ Square roots (miscellaneous) $\quad]$  [ Rational and irrational numbers ]

Find all values of $a$ for which the expressions $a+\sqrt{15}$ and $1 / a-\sqrt{15}$ take integer values.
Let $a+\sqrt{15}=m, 1 / a-\sqrt{15}=n$, where $m$ and $n$ are integers. Then $(m-\sqrt{15})(n+\sqrt{15})=1 \Leftrightarrow(m-n) \sqrt{15}=$ $16-m n$. The numbers $16-m n$ and $m-n$ are integers. If $m \neq n$, then the number $16-m n / m-n$ is rational, so it cannot be equal to $\sqrt{15}$. If $m=n$, then $16-m n=0$, hence $m=n= \pm 4$.

## Answer

$a= \pm 4-\sqrt{15}$.