Question

number of prime factors of (3 x 5)12 (2 x 7)10 (10)25 is:

Answer 1

The equation can be facorize as 3*5*3*2*2*2*7*2*5*2*5*5*5 or 2^5*3^2*5^5*7^1 total no of prime factor =(5+1)*(5+1)*(2+1)*(1+1)=216

Answer 2

Express the given number N as a product of prime numbers:

N
= $(3 * 5)12 (2 * 7)10 (10)25=3*5*2^2*3*2*7*2*5*2*5*5*5\\ \\=2^5*3^2*5^5*7$

A prime factor of N is obtained by multiplying  (none or one or two or three or four or five of 2s ) * (none or one or two of 3s ) * (none or one or more of 5s ) * (none or one 7 )

So the number of distinct prime factors will be 

         = (1+5)(1+2)(1+5)(1+1)  = 6 * 3 * 6 * 2  
 
       = 216