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:

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


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


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



Sources