A number theory problem by أحمد الحلاق

Say for example, we take the maximum amount of $2s$ that fit into $213$ . That is $212/2$ or maximum $106$ .

Now we say that we take the maximum amount of $13s$ that fit into $213$ . That is $208/13$ or maximum $16$ .

Evidently, to achieve the minimum amount of $m$ and $n$ , we need the amounts of 13s, or $n$ , to be as big as possible, and also be an odd number. Since odd x odd = odd, the maximum is $13$ x $15 = 195$ . Hence $n = 15$

Subtracting, $213 - 195 = 18$ . $18/2 = 9$ , and hence $m = 9$

Smallest value then is $m + n = 24$