Exercise 749Number theory › Divisibility and gcdHigh schoolAdded 2 Nov 2026

When is $(6n+12)/(2n+1)$ an integer?

#divisibility#fraction#case analysis

Statement

Find all integers $n$ such that $\dfrac{6n+12}{2n+1}$ is an integer.

Hint

The fraction is an integer iff $2n+1\mid 6n+12$ . Reduce: $6n+12=3(2n+1)+9$ .

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Reduction.
$6n+12=3(2n+1)+9.$
So $2n+1\mid 6n+12$ iff $2n+1\mid 9$ .

Case analysis. The divisors of $9$ are $\pm1,\pm3,\pm9$ . Hence
$2n+1\in\{-9,-3,-1,1,3,9\}.$
Solving for $n$ :
$n\in\{-5,-2,-1,0,1,4\}.$

Remark. This is a standard exercise from a Terminale Spécialité devoir.