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Problem 1. Find all pairs positive integer numbers $(m,n)$ such that $m+n^2$ is divisible by $m^2-n$ and $n+m^2$ is divisible by $n^2-m$

Problem 2. Find all positive integer numbers $p,q,r$ such that $1<p<q<r$ and $pqr-1$ is divisible by $(p-1)(q-1)(r-1)$

Problem 3. Find all positive integer numbers $(a,b)$ such that $a^2-2$ is divisible by $ab+2$

Problem 4. Find all positive integer numbers $(a,b)$ such that $a^2b+a+b$ is divisible by $ab^2+b+7$

Problem 5. Find all positive integer numbers $(a,b)$ such that $a+b^2a$ is divisible by $a^2b-1$

Problem 6. Find all positive integer numbers $(a,b)$ such that $a^3+1$ is divisible by $ab-1$

Problem 7. Find all positive integer numbers $(a,b)$ such that $a^2+b^2+1$ is divisible by $ab$

Problem 8. Find all positive integer numbers $(a,b)$ , $1\leq a\leq b$ such that $a^2+1$ is divisible by $b$ and $b^2+1$ is divisible by $a$

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