Casting Out Nines (and Elevens and Sevens . . .)

Most of you probably know (or knew at one time) the trick from pre-calculator days for deciding quickly whether a number is divisible by 9. You add up all the digits, and if the result is divisible by 9, then so was the original number. If you like, you can repeat the process, just summing the digits until only a one-digit number remains; if you started with a multiple of 9, the number you get will be a 9; otherwise, it won’t. I’ve heard that this process is well-known to numerologists under the name “casting out nines”.

What you may not know is that the same rule tests for divisibility by 3, and that a very similar rule tests for divisibility by 11.

Direct Digit Casting for 3 and 9: Start with a number. Add up the digits to get a new number. (Repeat as desired.) The original number is divisible by 3 (or 9) if and only if the new number is divisible by 3 (or 9).

Alternate Digit Casting for 11: Start with a number. Working from right to left, alternately add and subtract the digits. (Repeat as desired.) The original number is divisible by 11 if and only if the new number is divisible by 11.

Examples:

is divisible by 9 because is divisible by 9.

is divisible by 3 but not by 9, because is divisible by 3 but not by 9.

is not divisible by 3, because is not divisible by 3.

is divisible by 11 because is divisible by 11.

is not divisible by 11 because is not divisible by 11.

So what is going on? Why does this work? Are there other rules like this?

Most of you probably know (or knew at one time) the trick from pre-calculator days for deciding quickly whether a number is divisible by 9. You add up all the digits, and if the result is divisible by 9, then so was the original number. If you like, you can repeat the process, just summing the digits until only a one-digit number remains; if you started with a multiple of 9, the number you get will be a 9; otherwise, it won’t. I’ve heard that this process is well-known to numerologists under the name “casting out nines”.

What you may not know is that the same rule tests for divisibility by 3, and that a very similar rule tests for divisibility by 11.

Direct Digit Casting for 3 and 9: Start with a number. Add up the digits to get a new number. (Repeat as desired.) The original number is divisible by 3 (or 9) if and only if the new number is divisible by 3 (or 9).

Alternate Digit Casting for 11: Start with a number. Working from right to left, alternately add and subtract the digits. (Repeat as desired.) The original number is divisible by 11 if and only if the new number is divisible by 11.

Examples

• is divisible by 9 because is divisible by 9.
• is divisible by 3 but not by 9, because is divisible by 3 but not by 9.
• is not divisible by 3, because is not divisible by 3.
• is divisible by 11 because is divisible by 11.
• is not divisible by 11 because is not divisible by 11.

So what is going on? Why does this work? Are there other rules like this?