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If x is a positive integer such that. $2x + 12$ is perfectly divisibly by $x$, then the number of possible values of $x$ is:

1. $2$
2. $5$
3. $6$
4. $12$
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If (2x + 12) is perfectly divisible by x, then (2x + 12)/x has to be an integer as x is an integer.

Now if we divide, the expression simplifies to (2 + 12/x). The only way in which this expression would be an integer is when 12/x is an integer or if 12 is perfectly divisible by x.

This is possible if x takes either of these values : 1, 2, 3, 4, 6, 12.

Hence the answer is 6 values.