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Casting Out Nines - Math Trick

Figure out the missing digit in a large product of two integers. Echague Isabela

Effect :  The magician hands out a 3 or 4-digit integer chosen by a spectator in a previous part of the show.  
Using a pocket calculator, another spectator multiplies that number by some secret 3-digit number which he chooses freely and keeps for himself.  The result is a 6 or 7-digit number.  The spectator withholds one of those digits and reveals all the others in a random order.  The magician then reveals the withheld digit!

The trick is based on arithmetic modulo 9, which is what underlies the process of casting out nines from an integer, which is very familiar to schoolchildren  (at least it used to be).  Casting out nines  is a quick way to obtain the remainder when an integer  N  is divided by  9  (the key observation is that 10 and all the powers of 10 leave a remainder of 1, therefore, a number and the sum of its digits leave the same remainder).  (See elsewhere on this site for generalizations to other so-called  divisibility rules.)

Secret :  The number handed out by the magician is a multiple of 9  (Art Benjamin takes it from a random list of perfect squares; each of those has one chance in 3 of being divisible by 9 and the previous stage was not halted before a "good" number came up in that list).  The result is therefore a multiple of 9 and the sum of its digits is a multiple of 9.  When all the digits but one are revealed, the last one is thus known modulo 9.  This does reveal it unless it's either a zero or a nine.  In that ambiguous case, the magician will guess it to be a 9 and will almost always be right because people will rarely skip a zero when they are told to  skip any digit they like.Many variants of this trick can be devised based on any obscure process which produces a multiple of 9.

Here is one:

Ask a spectator to pick any 4-digit number and to consider the number obtained by reading it backwards.  Let the spectator secretly subtract the lesser number from the larger one, add 54 and multiply the result by any 3-digit number.  Ask how many digits there are in the final result and ask the spectator to keep one nonzero digit  secret and to reveal the other digits in scrambled order.  (Count ostensibly on your fingers how many digits you are given to make sure you're only missing one.)

You may then call the remaining digit with perfect accuracy.
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