Question

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

Answer 1

Note that: $$4!=2^3\cdot 3$$$$5!=2^3\cdot 3\cdot 5$$$$6!=2^4\cdot 3^2\cdot 5.$$Thus: $$4!5!6!=2^{10}\cdot 3^4\cdot 5^2=(2^2)^5\cdot (3^2)^2\cdot (5^2)^1.$$A perfect square is a number whose prime factorization has an even number of each prime thatt divides it. So the number of perfect squares that divide \(4!5!6!\) is \(6\cdot 3\cdot 2=36.\)