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:

 $\ds 12$ $=$ $\ds 4 \times \paren {1 + 2}$ $\ds$ $=$ $\ds 6 \times \paren {1 \times 2}$
 $\ds 24$ $=$ $\ds 4 \times \paren {2 + 4}$ $\ds$ $=$ $\ds 3 \times \paren {2 \times 4}$
 $\ds 36$ $=$ $\ds 4 \times \paren {3 + 6}$ $\ds$ $=$ $\ds 2 \times \paren {3 \times 6}$

It remains to be demonstrated that these are the only ones.

Let $z$ be a $2$-digit positive integer which is divisible by both the sum and product of its digits.

Neither of its digits may be $0$, because the product of the digits is a positive integer.

So, for $a \in \set {1, \dots, 9}$, $b \in \set {1, \dots, 9}$, $m \in \N$, $n \in \N$ we have:

 $\text {(1)}: \quad$ $\ds z$ $=$ $\ds 10 a + b$ $2$-digit positive integer. $\text {(2)}: \quad$ $\ds z$ $=$ $\ds \paren {a + b} m$ divisible by the sum of its digits $\text {(3)}: \quad$ $\ds z$ $=$ $\ds a b n$ divisible by the product of its digits

Consider $(1) = (2)$:

 $\ds 10 a + b$ $=$ $\ds \paren {a + b} m$ Note that if $m = 1$ then $a = 0$; hence $m > 1$ $\ds 10 a - a m$ $=$ $\ds b m - b$ rearranging $\ds a \paren {10 - m}$ $=$ $\ds b \paren {m - 1}$ factorizing $\text {(4)}: \quad$ $\ds \frac b a$ $=$ $\ds \frac {10 - m} {m - 1}$ rearranging; $a > 0$ and $m > 1$ so both denominators are never zero

In $(4)$ $a$, $b$, and $m - 1$ are all strictly positive; hence $10 - m$ must also be strictly positive.

Hence $1 < m < 10$.

Consider $(1) = (3)$:

 $\ds 10 a + b$ $=$ $\ds a b n$ $\ds n$ $=$ $\ds \frac {10 a + b} {a b}$ rearranging $\ds$ $=$ $\ds \frac {10 + \frac b a} b$ dividing numerator and denominator by $a > 0$ $\ds$ $=$ $\ds \frac {10 + \frac {10 - m} {m - 1} } b$ from $(4)$ $\ds$ $=$ $\ds \frac {10 \paren {m - 1} + 10 - m} {b \paren {m - 1} }$ multiplying numerator and denominator by $m - 1 > 0$ $\text {(5)}: \quad$ $\ds n$ $=$ $\ds \frac {9 m} {b \paren {m - 1} }$ simplifying

Since $n \in \N$, we must have:

$\paren {m - 1} \divides 9 m$

Since $m$ and $m - 1$ are coprime, by Euclid's Lemma:

$\paren {m - 1} \divides 9$

This gives $m \in \set {2, 4}$.

Consider the case $m = 2$:

From $(4)$, $\dfrac b a = 8$.

Hence the only potential value for $b$ is $8$.

From $(5)$, $n = \dfrac {18} 8 \notin \N$.

Hence there are no potential candidates for $z$ for the case $m = 2$.

Consider the case $m = 4$:

From $(4)$, $\dfrac b a = 2$.

Hence the only potential values are $b \in \set {2, 4, 6, 8}$.

From $(5)$, $n = \dfrac {12} b$.

However, $8$ does not divide $12$.

This leaves $b \in \set {2, 4, 6}$.

By virtue of $\dfrac b a = 2$, we have $z \in \set {12, 24, 36}$.

We have now considered all the potential values of $m$.

Hence we conclude that the only $2$-digit positive integers which are divisible by both the sum and product of their digits are $12$, $24$, and $36$.

$\blacksquare$