# 2-Digit Numbers divisible by both Product and Sum of Digits

## Theorem

The $2$-digit positive integers which are divisible by both the sum and product of their digits are:

$12, 24, 36$

## Proof

We have:

 $$\displaystyle 12$$ $$=$$ $$\displaystyle 4 \times \left({1 + 2}\right)$$ $\quad$ $\quad$ $$\displaystyle$$ $$=$$ $$\displaystyle 6 \times \left({1 \times 2}\right)$$ $\quad$ $\quad$

 $$\displaystyle 24$$ $$=$$ $$\displaystyle 4 \times \left({2 + 4}\right)$$ $\quad$ $\quad$ $$\displaystyle$$ $$=$$ $$\displaystyle 3 \times \left({2 \times 4}\right)$$ $\quad$ $\quad$

 $$\displaystyle 36$$ $$=$$ $$\displaystyle 4 \times \left({3 + 6}\right)$$ $\quad$ $\quad$ $$\displaystyle$$ $$=$$ $$\displaystyle 2 \times \left({3 \times 6}\right)$$ $\quad$ $\quad$