# 2-Digit Positive Integer equals Product plus Sum of Digits iff ends in 9

## Theorem

Let $n$ be a $2$-digit positive integer.

Then:

$n$ equals the sum added to the product of its digits
the last digit of $n$ is $9$.

## Proof

Let $n = 10 x + y$ where $0 < x \le 9, 0 \le y \le 9$.

Then:

 $\displaystyle \paren {x + y} + \paren {x y}$ $=$ $\displaystyle 10 x + y$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x y - 9 x$ $=$ $\displaystyle 0$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x \paren {y - 9}$ $=$ $\displaystyle 0$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle y$ $=$ $\displaystyle 9$ as $x \ne 0$

$\blacksquare$