If F is the product of all positive integers less than 31, than what is the greatest integer k for which F/18^k is an integer?  Pick one:
A) 3
B) 6
C) 7
D) 14
E) 26
30! F must be divisible by (3*3*2)^k.   As I will find a lot of 2's among 30!'s factors I will focus on the 3's (that also occurs twice instead of only once as a factor in 18).   Every multiple of 3 will provide me with atleast one factor of 3. There aren't that many so I list them quickly.   3 6 9*   12 15 18*   21 24 27**   30   The stars are marked as they are multiples of 9* or 27** and thus provides two or three 3's respectively.   3=3 (occurs 10 times) +10   3*3= 
<gadget id="calculator">3*3</gadget>
<output>9</output>
9 (occurs 3 times) +3(adding just one additional three each as they were not counted for in the multiples of 3)   3*3*3= 
<gadget id="calculator">3*3*3</gadget>
<output>27</output>
27 (occurs 1 time) +1 (again, adding just one as they were previously not counted)   Total: 14.   For every k we add two 3's. So we must divide 14 by 2 to get the highest integer K before we receive a non-integer result.   Answer: 7 -&gt; Answer choice: 7.
<result>C</result>
If F is the product of all positive integers less than 31, than what is the greatest integer k for which F/18^k is an integer?