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 $9$.
 * the last digit of $n$ is $9$.

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

Then: