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

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Theorem

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

Then:

$n$ equals the sum added to the product of its digits

if and only if:

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 $x \ne 0$

$\blacksquare$


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