Divisibility rule for 14

Answer 1

14 is difficult as it is 2 x 7 and 7 is difficult.

There are few, if any, divisibility rules which are quicker than dividing by 14, but one I can offer:

If the number is odd, it is not divisible by 14, otherwise apply the following test for divisibility by 7:

1. split the number into blocks of three digits from the right hand end (like normally writing a large number) - if a block has less than 3 digits, put leading zeros to make it 3 digits
2. Starting from the right hand end alternatively subtract and add the first digits of each block together, the second digits of each block together and the third digits of each block together.
3. Add twice the above sum of first digits to three times the sum of the second digits to the sum of the third digits
4. If the result of step 3 is divisible by 7, the original number is divisible by 7.

If the number is both even (divisible by 2) and divisible by 7 it will be divisible by 14.

Examples:

• 563361

Odd (not divisible by 2), so not divisible by 14.

• 74024562936

Even (divisible by 2), so apply 7 test:

74024562936 → 074 024 562 936

→ 2 x (9 - 5 + 0 - 0) + 3 x (3 - 6 + 2 - 7) + (6 - 2 + 4 - 4) = 2 x 4 + 3 x -8 + 4 = -12: not divisible by 7

⇒ 74024562936 is not divisible by 14

• 123789456

Even (divisible by 2), so apply 7 test:

123789456 → 123 789 456

→ 2 x (4 - 7 + 1) + 3 x (5 - 8 + 2) + (6 - 9 + 3) = 2 x -2 + 3 x -1 + 0 = -7: divisible by 7

⇒ 123789456 is divisible by 14