Question

A number N when factorised can be written $N=a^4\times b^3\times c^7$. Find the number of perfect squares which are factors of N. (The three prime numbers a, b, c > 2)

Answer 1

In order that the perfect square divides $N$ the powers of ${}^{\prime }{a}^{\prime }$ can be $0,2or4i.e.3$ Powers of ${}^{\prime }{b}^{\prime }$ can be $0,2,i.e.2$ Power of ${}^{\prime }{c}^{\prime }$ can be $0,2,4or6i.e.4$
Hence a combination of these powers given $3\times 24i.e.24$ numbers
so there are $24$ perfect squares that divides N