SOLUTION: The LCM of 6, 12, and n is 660. Find all the possible values of n.

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Question 831380: The LCM of 6, 12, and n is 660. Find all the possible values of n.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
660=10%2A66=10%2A6%2A11=5%2A2%2A2%2A3%2A11=2%5E2%2A3%2A5%2A11
so the prime factorization is 660=2%5E2%2A3%2A5%2A11
Since the prime factorization for 6 and 12 are
6=2%2A3 and 12=2%5E2%2A3 ,
the factors 5 and 11 must come from n .
There could be other factors of n included in 2%5E2%2A3%2A5%2A11 .
The prime factorization of n could have
2 with exponents 0, 1, or 2,
and/or 3 with exponents 0 or 1,
in addition to 5 and 11.
That gives us 3%2A2=6 possible values for n :
2%5E0%2A3%5E0%2A5%2A11=highlight%2855%29
2%5E0%2A3%5E1%2A5%2A11=highlight%28165%29
2%5E1%2A3%5E0%2A5%2A11=highlight%28110%29
2%5E1%2A3%5E1%2A5%2A11=highlight%28330%29
2%5E2%2A3%5E0%2A5%2A11=highlight%28220%29
2%5E2%2A3%5E1%2A5%2A11=highlight%28660%29