j_shreyans wrote:The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?
A)3
B)14
C)30
D)42
E)70
The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3.
Draw a VENN DIGRAM showing where the prime factors of n, 16 and 45 overlap:
The diagram implies the following:
Since n and 16 have only two 2's in common, the prime-factorization of n includes exactly two 2's.
Since n and 45 have only a 3 in common, the prime-factorization of n includes exactly one 3 and no 5's.
Thus:
n = 2*2*3*k, where k is not a multiple of 2, 3, or 5.
Since n is not a multiple of 5, the GCF of n and 210 cannot be a multiple of 5.
Eliminate C and E.
Since n = 2*
2*3*k and 210 =
2*3*5*7, the GCF of n and 210 must be a multiple of 6, as indicated by the factors in red.
Eliminate A and B.
The correct answer is
D.
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