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

In a 3-digit number N, the digits are non-zero and distinct such that none of the digits is a perfect square, and only one of the digits is a prime number. Then, the number of factors of the minimum possible value of N is

If $m$ and $n$ are natural numbers such that $n > 1$, and $m^{n}=2^{25} \times 3^{40}$, then $m-n$ equals

The sum of the first two natural numbers, each having $15$ factors (including $1$ and the number itself), is

The number of positive integers less than $50$, having exactly two distinct factors other than $1$ and itself, is

Let $n$ be the least positive integer such that 168 is a factor of $1134^n$. If m is the least positive integer such that $1134^{n}$. is a factor of $168^m$, then $m + n$ equals

How many pairs $(a, b)$ of positive integers are there such that $a \le b$ and $ab =$ $4^{2017}$?

How many factors of $2^4 \times 3^5 \times 10^4$ are perfect squares which are greater than $1?$

If $N$ and $x$ are positive integers such that $N^{N}=2^{160}$ and $N^{2}+2 ^N$ is an integral multiple of $2^x$, then the largest possible $x$ is

If the product of three consecutive positive integers is $15600$ then the sum of the squares of these integers is