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Divisibility by 9 and 11

Earlier Mathematics of the Day pages have dealt with quick tests to see if a number is divisible by 2, 3, 4, 5, 6, 8, 10 and 12. Today we will fill in some of the gaps in this list and look at quick tests for multiples of 9 and 11


Multiples of 9

This time, the test is just like that for divisibility by 3: find the single digit sum. So add all the digits to find their sum. If that is not a single digit, repeat. The final single digit tells us the remainder when we divide the original number by 9. So,
only if the digit sum is 9 is the original number an exact multiple of 9
For example, 314159 has a digit sum of 3+1+4+1+5+9 which is 23. 2+3 has a digit sum of 5, and, since this single digit is not 9, then 314159 is not a multiple of 9.
Since the remainder on dividing 314159 by 9 is 5 (the digit sum), then if we add 4 to the number, its digit sum will be 9 and the number will be a multiple of 9. 314159+4=314163, so 314163 is a multiple of 9.

Here is an interactive panel for you to practice on:

Interactive Practice Panel for Divisibility by 9

a multiple of 9.


Multiples of 11

Starting from the right hand end of the number and moving to the left, take the first digit, subtract the next digit, add the one after that, subtract the next and so on, alternately subtracting and adding the digits. If the resulting number is a multiple of 11, so is the original number. For this test, we may end up with a negative number. If you do, just ignore the minus sign and treat it as if it was positive.
Tip: Two-digit numbers where both digits are the same are always multiples of 11, that is 11, 22, 33, 44, and so on up to 99.
  • If the final answer is 11 or 0, the original number is a multiple of 11.
  • If the final answer is bigger than 11, do the same on its digits or use the tip above.
For example, 811414. The arithmetic we perform is 4-1+4-1+1-8=-1.
Because this is a negative number we ignore the minus sign and are left with 1.
This is not a multiple of 11, and so neither is 811414.

Another example: 1080706:

6-0+7-0+8-0+1=22. 22 is a multiple of 11 so 1080706 is also.

And for our final example: 92939190:
0-9+1-9+3-9+2-9= -30. Ignore the minus sign and we have 30, which is not a multiple of 11, which means that 92939190 is not either.

Here is an interactive panel for you to practice on:

Interactive Practice Panel for Divisibility by 11

a multiple of 11.


What about a test for multiples of 7?

Martin Gardner mentions several tests for divisibility by 7 in his Further Mathematical Diversions, Penguin books, 1977, in chapter 14: Divisibility Tests but there are no tests as simple as the ones we have already looked at for 2, 3, 4, 5, 6, 8, 9, 10 or 11. This book is now out of print, but it is well worth searching for in your local library or second-hand book shop.


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Primes, Factors and Divisibility