Solution:

The number is 72, simply because
72 = 8 (7 + 2).

How does one find this number?

Let t be the digit in the tens place and u be the digit in the units place. Then, the number is 10 t + u, and the sum of its digits is t + u. The following equation can be readily written:

10 t + u = 8 t + 8 u or
2 t = 7 u.

Thus, t / u = 7 / 2. Since t and u are digits, t must be 7 and u must be 2.

Food for thought:

Is the assumption of a 2-digit number a valid one?