Question

The number of positive divisors of $2^3{3}^5{5}^7{7}^9$ which are not perfect squares is ____

• A. $120$
• B. $1800$
• C. $1200$
• D. $1720$

Answer 1

The correct option is B $1800$

Any divisor of the $2^3{3}^5{5}^7{7}^9$ will be of the form $2^a{3}^b{5}^c{7}^d$

If a divisor is a perfect square, then the powers of the prime factors will be even.

So possible values of $a=0,2$

possible values of $b=0,2,4$

possible values of $c=0,2,4,6$

possible values of $d=0,2,4,6,8$

So no. of divisors which are perfect square $=\left(2\right)\left(3\right)\left(4\right)\left(5\right)=120$

Total no of divisors $=\left(4\right)\left(6\right)\left(8\right)\left(10\right)=1920$

So no. of divisors which are not perfect square $=1920-120=1800$