# 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:

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

\(\displaystyle \) | \(=\) | \(\displaystyle 6 \times \left({1 \times 2}\right)\) | $\quad$ | $\quad$ |

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

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

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

\(\displaystyle \) | \(=\) | \(\displaystyle 2 \times \left({3 \times 6}\right)\) | $\quad$ | $\quad$ |

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $36$