SE-SKOLORNAS 2016 Problem 5

Peter wants to create a new multiplication table for the four numbers $1, 2, 3, 4$ in such a way that the product of two of them is also one of them. He wants also that $(a\cdot b)\cdot c = a\cdot (b\cdot c)$ holds and that $ab \ne ac$ and $ba \ne ca$ and $b \ne c$ . Peter is successful in constructing the new table. In his new table, $1\cdot 3 = 2$ and $2\cdot 2 = 4$ . What is the product $3\cdot 1$ according to Peter's table?