Write a positive decimal integer out digit by digit in the form . The following rules then determine if is divisible by another number by examining the Congruence properties of its digits.

- 2. , so for . Therefore, if the last digit is divisible by 2 (i.e., is Even), then so is .
- 3. , , , ..., (mod 3). Therefore, if is divisible by 3, so is .
- 4. , , ... (mod 4). So if the last two digits are divisible by 4, more specifically if is, then so is .
- 5. , so for . Therefore, if the last digit is divisible by 5 (i.e., is 5 or 0), then so is .
- 6. , , ..., (mod 6). Therefore, if is divisible by 6, so is . A simpler rule states that if is divisible by 3 and is Even, then is also divisible by 6.
- 7. , , , , , (mod 7), and the sequence then repeats. Therefore, if is divisible by 7, so is .
- 8. , , , ..., (mod 8). Therefore, if the last three digits are divisible by 8, more specifically if is, then so is .
- 9. , , , ..., (mod 9). Therefore, if is divisible by 9, so is .
- 10. (mod 10), so if the last digit is 0, then is divisible by 10.
- 11. , , , , ... (mod 11). Therefore, if is divisible by 11, then so is .
- 12. , , , ... (mod 12). Therefore, if is divisible by 12, then so is . Divisibility by 12 can also be checked by seeing if is divisible by 3 and 4.
- 13. , , , , , (mod 13), and the pattern repeats. Therefore, if is divisible by 13, so is .

**References**

Dickson, L. E. *History of the Theory of Numbers, Vol. 1: Divisibility and Primality.* New York: Chelsea,
pp. 337-346, 1952.

Gardner, M. Ch. 14 in *The Unexpected Hanging and Other Mathematical Diversions.* Chicago, IL: Chicago University
Press, 1991.

© 1996-9

1999-05-24