Problem Solving

What is the smallest positive integer n
for which 324 is a factor of 6^n ?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

OA is C

(6^n)/324 = (6^n)/(6^2*9)

Note that for choice A and B, we don't get a whole number integer as an answer so those can automatically be eliminated. However, for choice C, we see that we get 36/9 or 4. This choice makes the most sense.

324 can be factorized as 6^2 * 9

If n = 2 or 3 then value of 6^n will be smaller than 324. n = 4 fits perfectly.

So n = 4 is the answer.

grandh01 wrote:What is the smallest positive integer n
for which 324 is a factor of 6^n ?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

In my opinion the best way to deal with these types of problems is to prime factor them out. So...,

324=3*3*3*3*2*2 and 6=3*2
So now we can quickly see that 6^2=3*3*2*2 if we eliminate 3's and 2's from 324 we are left with 3*3. Now 324 must be a factor of 6^n so we must accommodate the left over 3's by "adding" two more 2, so we get 6^4 (because we need 2 more 6's for the 2 left over 3's)