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Question 27

$$n$$ is a positive integer such that $$12n$$ is divisible by $$12 + n$$, then the number of possible values of $$n$$ is ______.


Correct Answer: 7

Solution

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Put $$m = n + 12$$, so that $$12n = 12m - 144$$ and $$m$$ must divide 144. Since $$n \ge 1$$ we need $$m > 12$$, and the divisors of 144 greater than 12 are $$16, 18, 24, 36, 48, 72, 144$$. These give $$n = 4, 6, 12, 24, 36, 60, 132$$, so there are 7 possible values.

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