If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18
A. 10
B. 12
C. 14
D. 16
E. 18
IMO C
I'll give you two ways to solve this problem pick whichever you are comfortable with:
1. Write down all the factors of 3 from 1-30
3,6,9,12,15,18,21,24,27,30
Now count how many 3 go into each of the numbers and add them up
1+1+2+1+1+2+1+1+3+1 = 14
2. To find the max power of a prime p present in any factorial f simply find
f/p + f/(p^2) + ..+ f/(p^n)
where p^n < f
So in this case we need to find the value of 30/3+30/(3)^2+30/(3)^3
(we stop at 3^3 since 3^4 > 30)
10+3+1 = 14 approx
