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Suppose $a,b,c$ are three distinct natural numbers, such that $3ac = 8(a+b)$. Then, the smallest possible value of $3a+2b+c$ is

If $m$ and $n$ are integers such that $(m+2n)(2m+n) = 27$, then the maximum possible value of $2m-3n$ is

The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x > y \geq 3$ and $x + y < 14$ is

Let $N, x$ and $y$ be positive integers such that $N = x + y,$ $2 < x < 10$ and $14 < y < 23$. If $N > 25$, then how many distinct values are possible for $N?$

The number of pairs of integers ( $x, y$ ) satisfy $x \geq y \geq-20$ and $2 x+5 y=99$ is

If $a, b$ and $c$ are positive integers such that $ab = 432, bc = 96$ and $c < 9,$ then the smallest possible value of $a + b + c$ is

How many pairs $(m, n)$ of positive integers satisfy the equation $m^2 + 105 = n^2$?

How many different pairs $(a, b)$ of positive integers are there such that $a ≤ b$ and $\frac{1}{a} + \frac{1}{b} = \frac{1}{9}$

The number of solutions $(x, y, z)$ to the equation $x - y - z = 25$, where $x, y$, and $z$ are positive integers such that $x \le 40, y \le 12$, and $z \le 12$ is