If we divide 12! from both each term, 12!/12! = 1, 15!/12! will have a factor of 3, 18!/12! will have a factor of 3, 21!/12! will have a factor of 3, and 24!/12! will also have a factor of 3.
So thie expression will turn into
12!(1 + some multiple of 3 + some multiple of 3 + some multiple of 3 + some multiple of 3)
1 + some multiple of 3 + some multiple of 3 + some multiple of 3 + some multiple of 3 = not a multiple of 3
So we have 12! * not a multiple of 3.
Therefore, the exponent of 3 of 12! + 15! + 18! + 21! + 24! is the same as the exponent of 3 for 12!.
12! = 1*2*3*4*5*6*7*8*9*10*11*12, (3, 6, 9, 12) are our numbers divisable by 3.
3 = 3
6 = 2*3
9 = 2*3^2
12 = 2^2*3
Therefore, 1 + 1 + 2 + 1 = 5 is the exponent of 3.
I'm sorry if the explanation is confusing, please ask if there are any questions.
=^._.^=
