## The answer in the problem

A PuzzleCritic Original:

Let x be the answer to this problem, where x is a positive integer, and let y be the sum of its digits. Calculate 2x-2y.

Since the answer to this problem is x and we are asked to calculate 2x-2y, it follows that

2x – 2y = x,

so that x = 2y. No single digit positive integer is double the sum of its digits, and for numbers with three digits or more, the sum of the digits is at most 9k (where k is the number of digits), but the number itself is at least $10^{k-1}$, which is much bigger than 9k for $k \geq 3$. So our only hope is to look for a two digit number.

If the digits are a and b, then x = 10a + b, and y = a + b, so we have

10a + b = 2a + 2b,

and thus

8a = b.

Since a and b are digits, the only possibility is a = 1 and b = 8, so that x = 18, y = 9, and

2x – 2y = 18.