SOLUTION: 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

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Question 165477: 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
Found 2 solutions by MRperkins, Edwin McCravy:
Answer by MRperkins(300) About Me  (Show Source):
You can put this solution on YOUR website!
Factor 324 completely: 324=2%2A2%2A3%2A3%2A3%2A3
6%5En
6%5E1=%282%2A3%29
6%5E2=%282%2A3%29%2A%282%2A3%29
Continue doing this until you have all of the factors of 324. It is ok to have additional factors but it is not ok to be missing any factor.
6%5E3=%282%2A3%29%2A%282%2A3%29%2A%282%2A3%29 notice that we are still missing a 3 in this one
6%5E4=%282%2A3%29%2A%282%2A3%29%2A%282%2A3%29%2A%282%2A3%29 here we finally have two (2)'s and the four (3)'s that are in 324 and we also have two (2)'s left over.
6^4 =324(the factor of 6^4) times the two (2)'s that are left over
so 6^4=1296
Edwin is correct that (c)4 is the right answer.

Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
Edwin's solution:
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 

Break 324 down into primes

324 = 2*2*3*3*3*3

Since 6 = 2*3, each factor of 6 contributes one factor of 2 
and one factor of 3.

Since 324 has four factors of 3, we need four factor of 6 to
contribute them all.

So the answer is 4, choice (C)

Edwin