­If the greatest common factor of the integers n/11 and n/13 is 6

Official Solution:

If the greatest common factor of the integers \(\frac{n}{11}\) and \(\frac{n}{13}\) is 6, what is the number of positive factors of the positive integer \(n\)?

A. 2
B. 4
C. 6
D. 8
E. 16


\(n\) must be a multiple of 2 and 3. Because \(6=2*3\) is a factor of \(\frac{n}{11}\) and \(\frac{n}{13}\), then \(6=2*3\) must also be a factor of \(n\). Additionally, notice that \(n\) cannot have higher powers of 2 or 3 because in this case the GCF would contain higher powers of these primes;

\(n\) must be a multiple of 11 because we are told that \(\frac{n}{11}\) is an integer. \(n\) cannot have a higher power of 11 because in this case the GCF would contain 11;

\(n\) must be a multiple of 13 because we are told that \(\frac{n}{13}\) is an integer. \(n\) cannot have a higher power of 13 because in this case the GCF would contain 13;

\(n\) cannot be a multiple of any other prime because in this case that prime would also appear in \(\frac{n}{11}\) and \(\frac{n}{13}\) and thus in the GCF.

Thus, \(n\) must be \(2*3*11*13\). Therefore, \(n\) has \((1+1)(1+1)(1+1)(1+1)=16\) positive factors.


Answer: E­