CAT 2025 Slot 2 QA Question 17: The number of divisors of $2^6 imes 3^5 imes 5^3 imes 7^2$, which are of the form $3r+1$, where $r$ is a non-negative integer, is

Question

The number of divisors of $2^6 	imes 3^5 	imes 5^3 	imes 7^2$, which are of the form $3r+1$, where $r$ is a non-negative integer, is

Accepted Answer

A divisor has the form
$2^a 	imes 3^b 	imes 5^c 	imes 7^d$
where $0 \leq a \leq 6, 0 \leq b \leq 5, 0 \leq c \leq 3, 0 \leq d \leq 2$.

A number of the form $3r + 1$ is not divisible by $3$. Hence $b = 0$.

Now work modulo $3$.
Modulo $3$, $2 \equiv -1$, $5 \equiv -1$ and $7 \equiv 1$.
So the div

The number of divisors of 26×35×53×722^6 \times 3^5 \times 5^3 \times 7^226×35×53×72, which are of the form 3r+13r+13r+1, where rrr is a non-negative integer, is