24 = 2*2*2*3.nahid078 wrote:If m is a positive integer, is m/24 also an integer ?
(1) GCF of m and 40 is 4
(2) GCF of m and 84 is 12.
For m to be divisible by 24, the prime-factorization of m must include AT LEAST THREE 2's AND AT LEAST ONE 3.
Question stem, rephrased:
Does the prime-factorization of m include at least three 2's and at least one 3?
GCF = greatest common factor.
Statement 1: GCF of m and 40 is 4
40 = 2*2*2*5.
Since the GCF of m and 40 includes only the factors in red, none of the factors in green are included in the the prime-factorization of m.
Thus, the prime-factorization of m CANNOT include at least three 2's, and the answer to the question stem is NO.
SUFFICIENT.
Statement 2: GCF of m and 84 is 12
84 = 2*2*3*7.
Since the GCF of m and 84 is 12, all of the factors in green are included in the prime-factorization of m.
Only the factor in red -- 7 -- may not be included in the prime-factorization of m.
Case 1: m = 2*2*3
Here, the prime-factorization of m does NOT include at least three 2's and one 3, so the answer to the question stem is NO.
Case 2: m = 2*2*2*3
Here, the prime-factorization of m DOES include at least three 2's and one 3, so the answer to the question stem is YES.
Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.
The correct answer is A.
