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

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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:

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


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


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



Sources