Maths question.?

Question

This is just a puzzle, I know the answer but just for general amusement or annoyance.

When you take any three digit number, not a reverse of itself (e.g 787).
Reverse it and subtract the smaller of the two from the larger.
(if left with a two digit number a zero must be placed in front of it, e.g 098)
Take the answer to this and its reverse and add them together.

Show why, in anyway you choose, why the answer will always be 1089.

E.G
765
-567
- - -
198
+891
- - -
1089

(A university lecturer did this trick on me and I was annoyed for the next few days trying to work out why.)

Answer 1

if the digits in 3-digit number is a, y, z
So the number is = 100x + 10y + z
Reverse is = 100z + 10y -x
Subtract second number from 1

|(100x + 10y + z) - (100z + 10y -x)| = 99|(x - z)|

The quantity x - z lies between 2 and 9 and the difference is a 3-digit multiple of 99.
So the numbers are 198, 297, 396, 495, 594, 693, 792 or 891.
Observation:
a. The middle digit is always 9.
b. The sum of 1 and 3rd digit is always 99.

Therefore adding the number and its reverse is always 1089.