Divisibility Rules

Contributed by Mickey Sheu of Naperville North High School. Edited by Ian Yap and Abe Hassan and Scott Halvorson

Divisibility By Seven

  1. Take the last digit off of the number.
  2. Double this digit.
  3. Subtract this new number from the rest of the original number.
  4. Keep repeating until you get to a point when you cannot go further.
  5. If this number is divisible by seven, then so is the original number.

Example:

  1. Number: 12345
  2. Take off 5, double it to get 10, and subtract: 1234 - 10
  3. Result: 1224
  4. Take off 4, double it to get 8, and subtract: 122 - 8
  5. Result: 114
  6. Take off 4, double it to get 8, and subtract: 11 - 4
  7. Result: 3
  8. Since three is obviously indivisible by seven, 12345 is also not divisible by seven.

Proof: All you are doing is subtracting 21*the last digit, preserving the remainder.

Divisibility By Eleven

  1. Take the last digit off of the number.
  2. Subtract this number from the rest of the original number.
  3. Keep repeating until you get to a point when you cannot go further.
  4. If this number is divisible by eleven, then so is the original number.

Example:

  1. Number: 12345
  2. Take off 5 and subtract: 1234 - 5
  3. Result: 1229
  4. Take off 9 and subtract: 122 - 9
  5. Result: 113
  6. Take off 3 and subtract: 11 - 3
  7. Result: 8
  8. Since eight is obviously indivisible by eleven, 12345 is also not divisible by eleven.

Proof: All you are doing is subtracting 11*the last digit, preserving the remainder.

And now, an addition to Divisibility by Eleven (an alternate method):

  1. Add and subtract every other digit, going from left to right.

  2. If the end result is divisible by 11, then so is the original number.

Example:

  1. Number: 36,948,216,437

  2. Add and subtract: 3 - 6 + 9 - 4 + 8 - 2 + 1 - 6 + 4 - 3 + 7 = 11

  3. Since 11 is obviously divisible by 11, then so is 36,948,216,437 (3,358,928,767 * 11).