How many perfect squares divide the number 4!·5!·6! ?

Question

geerky42 · Answer

How did you get that answer?

anonymous · Answer

take that back hold on.....what else did it say

geerky42 · Answer

Nothing else.

anonymous · Answer

is it like 4*5*6=120

geerky42 · Answer

120 is a reasonable answer, but your solution doesn't make sense. It is asking for number of perfect squares that 4!·5!·6! are divisible of.

anonymous · Answer

You'll want to prime factor the number first:$$4!\cdot 5!\cdot 6!=(4!)^3\cdot 5^2\cdot 6=24^3\cdot 5^2\cdot 6=(3\cdot 2^3)^3\cdot  5^2\cdot (2\cdot 3)$$$$=2^{10}\cdot 3^4\cdot 5^2$$

anonymous · Answer

Now start counting the ways you can factor squares out.  Its a little tedious (unless there is a clever counting argument available, im not good with combinatorics). Just looking at the powers of 2, its clear that:$$2^2,2^4,2^6,2^8,2^{10}$$are all squares that divide your numbers.

anonymous · Answer

Maybe this will work, factor the primes as such:$$(2^2)^5\cdot (3^2)^2\cdot (5^2)$$Then its clear that the number of square divisors is:$$(5+1)\cdot (2+1)\cdot (1+1)=36$$

geerky42 · Answer

Smart. Thanks!