mathschallenge.net

Problem

The number 49 is unusual in that 49 + (4+9) = 36 + 13 = 49. In other words, if we add the product and sum of the digits we get the number itself.

Find all 2-digit solutions.

Solution

Let the number, N = 10$a$+$b$ = $ab$ + $a$+$b$.

9$a$ = $ab$, 9$a$ $ab$ = 0, $a$(9 $b$) = 0 $a$ = 0 or $b$ = 9.

But as N is a 2-digit number, $a$ 0.

Hence the 2-digit solutions are: 19, 29, 39, ..., 99.

What about 3-digit solutions?