SOLUTION: Show the difference between a four-digit number and its reverse is divisible by 9

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Question 148905: Show the difference between a four-digit number and its reverse is divisible by 9
Answer by Edwin McCravy(20056) About Me  (Show Source):
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Show the difference between a four-digit number and its reverse is divisible by 9 

Suppose the number is "ABCD".

Then the value of the number is 

1000A + 100B + 10C + D

and its reverse is "DCBA" with value

1000D + 100C + 10B + A

Now suppose "ABCD" is larger than "DCBA", then if we subtract them,

(1000A + 100B + 10C + D) - (1000D + 100C + 10B + A)

Remove the parentheses:

1000A + 100B + 10C + D - 1000D - 100C - 10B - A

999A + 90B - 90C - 999D
 
Factor out 9

9(111A + 10B - 10C - 111D)

This is a multiple of 9.

If "DCBA" is larger than "ABCD" we'll end up with

9(111D + 10C - 10B - 111A) 

This is also a multiple of 9.

Edwin