soumya1989 wrote:

If \(N = 3^x*5^y\), where x and y are positive integers, and N has 12 positive factors, what is the value of N?

Column A | Column B |

9 is not a factor | 125 is a factor |

A. Only Column A is sufficient.

B. Only Column B is sufficient.

C. Column A and B both are needed

D. Column A and B both are insufficient

E. None of the columns are needed.

For me, the ans would be A, we are told that \(N = 3^x*5^y\), where x and y are positive integers, and N has 12 positive factors.

then (x+1)(y+1)=12. Let us do some prime factorization of 12.

12=2*6=1*12=3*4 We should rule out the pair 1*12 since x+1 or y+1=1, then x or y=0. Yet, we are told that x and y are positive integers. Now for cloumn A, we are told that 9 is not a factor of N. This leads us to know x=1 and y=5. then \(N = 3^1*5^5\).

For column B, we are told that 125 is a factor. That means that y>=3. So, y would be 3 or 5. B can't help us decide the value of N.

ANS:A