## Find All Integers

Find all integers $n$ such that $(5n+23)/(n-7)$ is also an integer.
Source: NCTM Mathematics Teacher, December 2005

SOLUTION
$(5n+23)/(n-7)=5+58/(n-7)$
The expression $(5n+23)/(n-7)$ is an integer if $n-7$ is a divisor of $58$.
$58=1\cdot 2\cdot 29$
The divisors of $58$ are: $\pm 1,\pm 2,\pm 29,\pm 58$
$n-7=1, n=8$
$n-7=\textrm{-}1, n=6$
$n-7=2, n=9$
$n-7=\textrm{-}2, n=5$
$n-7=29, n=36$
$n-7=\textrm{-}29, n=\textrm{-}22$
$n-7=58, n=65$
$n-7=\textrm{-}58, n=\textrm{-}51$
Eight integers $\left \{\textrm{-}51,\textrm{-}22,5,6,8,9,36,65\right \}$ make the expression $(5n+23)/(n-7)$ also an integer.