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My dog takes the AIME, and sees a problem. The question's answer was $\frac{m}{n}$ and asked for $m+n$, where $m$ and $n$ are relatively prime positive integers. She randomly …

What is the smallest positive integer with exactly 17 positive divisors?

Let $a$ and $b$ be positive integers such that the number $b^2+(b+1)^2+(b+2)^2+\cdots+(b+a)^2-3$ is a multiple of $5$ and $a+b$ is odd. What is the units digit of the number $a+b$ …

Find the smallest positive integer $n \geq 10$ such that $n+6$ is a prime and $9n+7$ is a perfect square.

$x^y-y^x=xy^2-19$

Solve the equation above for primes $x$ and $y$. Given that there are two solution pairs $(x_1, y_1)$ and $(x_2, y_2)$, find the value of $(x_1+y_1)(x_2 + y_2)$.

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