Magic on the fly

Okay, I have to print the first four entries and bring them to class, and class meets in an hour, so I'll use one of my lifelines:  I'm writing a piece of my own class prep today  :)

So you are doing this magic show.  Ask for a volunteer from the audience, and ask her to think of a (whole) number between 1 and 99.  Running example:  Let's say she thinks of 59.

Now ask her to divide the number by 9 and tell you the remainder.  If she chose 59, she'll say "5".

Next, ask her to divide the number by 11 and tell you the remainder.  If she chose 59, she'll say "4".

You're going to amaze your audience by guessing the number she chose from the two remainders.  Here's how:

Add the two remainders together, multiply the answer by 10, then add the remainder from 9 and subtract the remainder from 11.  In the example:  You added the 5 and the 4, which gave you a 9; then you multiplied by 10, so you got to 90; then you added the 5 and got 95; then you subtracted the 4, so you have 91.

After the above, you don't even need the remainders any longer  :)  Just multiply what you have by 5---in the example, multiply the 91 by 5, which gives you 455---and if it's a 3-digit number, add the digit in the 100's place to the rest of it---in the example, add the 4 to the 55---and you have the answer.

(And if you get a 1-digit or 2-digit number in place of the 455, that would be the answer already.  Exception:  If you get a 0 in place of the 455, she chose 99.)

One more example:  Let's say your volunteer chooses 98.  She tells you that the remainder from 9 is 8, and that the remainder from 11 is 10.

You add the remainders and get 18; you multiply by 10 and get 180; you add the remainder from 9 and get 188; you subtract the remainder from 11 and get 178.  Now you can forget the remainders.  You multiply the 178 by 5 and get 890 (btw, this is the biggest number you can get at this stage); you add the 8 to the 90 and you have the 98.

Want to know (a) whether this really works for all whole numbers between 1 and 99, (b) why this works, (c) how to make up more magic tricks to depress your friends?  Talk to your local college about taking a number theory class!

And I'm actually done with this entry in time  :)