Problem Solving

[GMAT math practice question]

If n, n/3 and n/4 are positive integers, and n is less than or equal to 100, how many values of n are possible?

A. 6
B. 7
C. 8
D. 9
E. 10

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The condition n/3 is a positive integer tells us that n is a positive multiple of 3.
The condition n/4 is a positive integer tells us that n is a positive multiple of 4.
Thus, n is a positive multiple of 12.

The number of positive multiples of 12 less than or equal to 100 is 8 since 100 = 12*8 + 4.

Therefore, the answer is C.
Answer: C

$$n=1,2,3,---------------,100$$
$$\frac{n}{3}=all\ values\ of\ n\ which\ are\ divisible\ by\ 3$$
3,6,9,12,15,18,21,24,27,30,33,36,39,42,45.48,51,54,57,60,63,66,69,72,75,78,81,84,87,90,93,96,99

$$\frac{n}{4}=all\ values\ of\ n\ which\ are\ divisible\ by\ 4$$
4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,96,100
$$look\ for\ \left\{E\left(\frac{n}{4}\right)C\left(\frac{n}{3}\right)C\left(n\right)\right\}$$

Elements in n/4 that belongs to n/3 and definitely will be found in
n = 12, 24, 36, 48, 60, 72, 84,96 This is a total of 8 elements.
8 values of n are positive.

$$answer\ is\ Option\ C\ $$