Give the following instructions to a friend:
Write down a three-digit number,
where the first and and last digits are different.
Reverse the order of the digits.
Subtract the smaller number from the larger.
Multiply the result by any number that you wish (Yes, any number!).
Now strike out any one of the digits in the answer (except a zero).
Now have your friend add the remaining digits and tell you the sum.
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You will now tell your friend the digit that he or she struck out.
(1) If the sum is less than 9,
subtract the sum from 9 and that is the digit struck out.
(2) If the sum is 9, then your friend struck out a 9.
(3) If the sum is more than 9, then add the digits of the sum together.
If the sum is still greater than 9,
keep adding the digits together until you get a single digit.
Then subtract from 9 to get the digit struck out.
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Why does this work?
In step one, when you subtract the numbers, you always get a multiple of 9.
That means that the sum of the digits is a multiple of 9.
When you multiply by any number, the answer will still be a multiple of 9.
So, the sum of the digits is a multiple of 9.