Question 94
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ that satisfy the following conditions:
* $f(a, b) + a + b = f(a, 1) + f(1, b) + ab$ holds for all positive integers $a$ and $b$.
* If any number among $a + b$ and $a + b - 1$ is divisible by prime number $p \gt 2$, then $f(a, b)$ is divisible by $p$ as well. (Ivan Krijan)
* $f(a, b) + a + b = f(a, 1) + f(1, b) + ab$ holds for all positive integers $a$ and $b$.
* If any number among $a + b$ and $a + b - 1$ is divisible by prime number $p \gt 2$, then $f(a, b)$ is divisible by $p$ as well. (Ivan Krijan)