Question

Given a number $N=4200$ , then the number of divisor of $N$ which are neither divisible by $3$ nor by $5$ is

Answer 1

$4200=42\times 100=\left(2 \times 3 \times 7\right)\left(2^{2} \times 5^{2}\right)$
$\Rightarrow 4200=2^{3}\cdot 3^{1}\cdot 5^{2}\cdot 7^{1}$
If divisor is neither divisible by $3$ nor by $5$ , then it should not have any power of $3$ as well as $5$ .
Number of divisors $=\left(3 + 1\right)\left(1\right)\left(1\right)\left(1 + 1\right)=8$