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.

Answer: 8

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About mvtrinh

Retired high school math teacher.
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