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 = 10a+b = ab + a+b.

9a = ab, 9a 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?