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

Question

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

Bunuel · Accepted Answer

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

gmatophobia · Answer

n = 11 * 13 * 6

n = 11^1 * 13^1 * 2^1 * 3 ^1

Number of factors  = 2 * 2 * 2 * 2 = 16

Option E

envy5 · Answer

I followed your reasoning until the second step but what is that last step? I was under the impression that the factors would be 1, 2, 3, 11, 13, and the number itself, so 6, but understand now that 33, 39 could be a factor etc.

gmatophobia · Answer

envy5  - You can refer to the below article on the method to find the number of positive factors of an integer

Finding the Number of Factors of an Integer

First make prime factorization of an integer  n=a^p*b^q*c^r , where  a ,  b , and  c  are prime factors of  n  and  p ,  q , and  r  are their powers.

The number of factors of  n  will be expressed by the formula  (p+1)(q+1)(r+1) .  NOTE:  this will include 1 and n itself.

Example:  Finding the number of all factors of 450:  450=2^1*3^2*5^2

Total number of factors of 450 including 1 and 450 itself is  (1+1)*(2+1)*(2+1)=2*3*3=18  factors.

Reference -  https://gmatclub.com/forum/math-number-theory-88376.html

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

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