Q:

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?

A:

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).

