A middle school teacher who read my recent blog on casting out nines wrote to ask a couple of questions. She was unable to answer a student question about why the casting out nine process gives the remainder when a number is divided by nine and wondered if I could explain. She went on to ask if there was a way to do something similar with eleven since she seemed to remember something about that in a workshop she saw some years ago. The answer to both questions is yes.

The secret behind casting out nines is hidden in the following set of equalities:1

10 = 1*9+1

100 = 11*9 +1

1000= 111*9+1

10000= 1111*9+1

So if we have a number like 1234, which is 1000+200 + 30 + 4, we can rewrite it as 1(111*9 + 1) + 2(11*9+1) + 3(1*9+1) + 4. Distributing the numbers in red gives me 111*9 + 1+ 22*9+2 + 3*9+3 + 4 All the blue terms have a multiple of nine, and so collecting by color we get 136*9 + 10. Since nine will divide evenly into the 136*9, the remainder is the remainder when ten is divided by nine, and of course we know that 10 = 1*9+1, so the remainder is one.

For elevens the pattern is slightly different. Notice that :

10 = 1*11 –1

100 = 9*11 +1

1000= 91*11-1

10000= 909*11+1

100,000= 9091*11-1

….. and each multiple of ten alternates being one more than or one less than a multiple of 11. So if we expand 1234 as before into 1000+200 + 30 + 4, we can rewrite it as 1(91*11 –1) + 2(9*11+1)+ 3(1*11-1) + 4 . Expanding as before we get some powers of eleven, and the remaining numbers are –1 + 2 – 3 + 4 = 2; so the 1234 divided by 11 leaves a remainder of 2.

It is very similar to the method of casting out nines, except that we alternate adding and subtracting instead of adding all the terms. Most people find it easier to start at the right since the constant term is always added, but you must be able to deal with small negative numbers. For instance if we use 4152, we will do 2 – 5 + 1 – 4 which yields –6. We know there can not be a negative remainder, so what to do. Simply remove this amount from the base, 11, and the answer is revealed. The remainder is 5.

This should remind you of the “clock arithmetic” you probably first learned for doing modular math. On a 12 hour clock, if we think of 2 – 4 (four hours before 2 o’clock) we get 10. 2-4 = -2 and 12 –2 = 10. Gosh I hope that is clear.