Pat the mathemagician Pat the Magician asks a member of the audience to write a 13-digit number on the blackboard, while Pat is blindfolded. Pat then asks the volunteer to reverse the digits of the number, and to subtract the smaller of the two 13-digit numbers from each other. (If the original number ended in one or more zeros, the reversed number of course will have less than 13 digits.) Then Pat asks the volunteer to circle any one of the digits in the difference, provided the digit is not zero. Then Pat asks the volunteer to say what the uncircled digits were. After hearing all of this, Pat is able to tell the amazed volunteer what the circled digit was (Pat is blindfolded during this entire trick). How does Pat do the trick?

Show that the difference of the 2 13-digit numbers, is a multiple of 9.

The rule of divisibility for 9 is that the sum of digits will be a multiple of 9. So if he knows all but one of the digits, he can find the last one (assuming no calculation error).