How to Quickly and Easily Tell if a Number is Divisible by 11, and Other Math Tricks

Daven Hiskey November 30, 2012 10
Now You KnowYou should know how to easily tell if a number is divisible by 11. (many more fun math tricks below)

As an example, we’ll use the number 10604.

  • First, add up the odd number digits in the number: 1 + 6 + 4 = 11.
  • Next, add up the even number digits: 0 + 0= 0.
  • Now subtract the sum of the odd digits (11) with the sum of the even digits (0): 11-0 = 11.
  • Now take the resulting number and see if you can divide it by 11: 11/11 = 1

If you can do this final division, as in this case (11/11 = 1), then the number itself (10,604) is also divisible by 11.

So that’s how to tell if something’s divisible by 11.  How about multiplying any 2 digit number by 11 easily in your head?

Simply take the 2 digit number, we’ll us 62, then

  • add a space holder in between, so 6_2.
  • Now add those 2 numbers together (6 + 2 = 8).
  • Now put the 8 in the space holder: 682 = 11*62.

Now I know what you’re thinking, what happens if the two numbers add up to greater than 9?  Do I simply make 2 spaces?  Nope. To see what to do here, we’ll use the number 79.

  • 7_9;
  • 7+9 = 16.
  • Now take the “ones” digit (6) and place it in the empty space: 769.
  • Now add the “tens” digit (1) to the number directly in front of the space, so in this case: 7+1 = 8: so the result is 869, which is 11*79. (Note: this still works even if the “tens” number added exceeds 9; for example: 99; 9_9; 9+9 = 18; 989; 9 + 1 = 10; 1089 = 11 * 99.

Now there’s also a way to do this multiplying any number by 11, but it’s slightly more complicated to do in your head, (super easy on paper, but if you’ve got pen and paper, there isn’t really so much of a need for a trick!)  This can still be done in your head though, but it’s going to sound a little complicated at first until you’ve practiced it a couple times.

All you’ve got to do is use an “add the neighbor trick”.  Take a number like  1,342.

  • Mentally add 0 in front of it, so 01,342.  Now simply start at the right and “add the neighbor”.
  • 2 has no neighbor to the right, so you just leave it in your head there (2).
  • 4′s neighbor is 2, so you add them together and get 6, so (62).
  • 3′s neighbor is 4, so add them together to get 7, so (762).
  • 1′s neighbor is 3, so add them together to get 4, so (4,762).
  • The 0′s neighbor is 1, so add them together to get 1, so (14,762).

That’s it: 11 * 1,342 = 14,762.

Bonus Math Tricks and Facts:

  • In order to easily square in your head any 2 digit number ending in a 5 (we’ll use 65 here), simply
    • add 1 to the “tens” digit, so 6+1 = 7.
    • Now multiply the original “tens” digit with the resulting number, so 6*7 = 42.
    • Now just put 25 after that number, so 4225.
    • Thus, 65 squared is 4225.
  • 111111111×111111111 = 12345678987654321
  • In a group of 23 people, there is about a 50% chance that 2 of the 23 will have the same birthday.
  • Everything you can mathematically do with a ruler and a compass you can do with the compass alone.
  • The equals sign (“=”) was invented in 1557 by Welsh mathematician Robert Recorde, who was fed up with writing “is equal to” in his equations. He chose the two lines because “no two things can be more equal”.  Recorde is also the one who introduced the plus and minus signs to Brittan, though he didn’t invent them.
  • If “z” is the radius and “a” is the height, the mathematical volume of a pizza is pi*z*z*a.
  • To easily tell if a number is divisible by 3 in your head, just check if the sum of all the digits in the number is divisible by 3.  If so, then the number itself is also divisible by 3.  For example, 387: 3+ 8 + 7 = 18.  18 /3 = 6.  Thus, 387 is divisible by 3.
  • Want to know if a number is easily divisible by 6?  Just check and see if it’s divisible by both 2 (if the last digit is even) and is divisible by 3 using the above trick.  If it is on both counts, then it is also divisible by 6.
  •  You can tell if a number is divisible by 8 by simply looking at the last 3 digits in the number and checking to see if they are divisible by 8.  If so, then the number itself is also divisible by 8.  For example, 129,846,104: 104 / 8 = 13, thus, 129,846,104 is divisible by 8.
  • A similar trick can be used to see if a number is divisible by 4.  Simply take the last 2 digits and check if they are divisible by 4.  If so, then the number is divisible by 4.  Thus, 628,834,221,912: 12 /4 = 3, so 628,834,221,912 is divisible by 4.
  • If you want to know if a number is divisible by 12, simply use the above tricks to see if it’s divisible by 3 and 4.  If it is divisible by both, then it’s also divisible by 12.
  • For checking if a number is divisible by 7, (we’ll use 224 as the example) simply
    • double the last digit in the number, 4*2 = 8
    • then subtract this from the rest of the number, 22-8 = 14
    • Now if the result is divisible by 7, ( 14 / 7 = 2), then the original number (224) is divisible by 7.
  • On this “7″ trick, if the resulting number is still too big to easily tell if it’s divisible by 7, simply perform the trick again (recursively) on the resulting number until you get down to a sufficiently small number that you can easily tell it’s divisible by 7.  For example, 2296: 6*2 = 12; 229 – 12 = 217.  Now is 217 divisible by 7?  Still might not be clear in your head.  So then do the operation again on 217:  7*2 = 14; 21 – 14 = 7; 7/7 = 1. So, yes, 2296 is divisible by 7.
  • Want a trick for dividing any number by 5 fairly easily? (Particularly for numbers not too large- it gets more complicated to do in your head when the numbers start to get really big.)  Simply take the number, we’ll use 412, and double it, so 824.  Now add a decimal point before the “ones” digit, so 82.4 = 412 /5.  An example with a slightly bigger number 1,024 * 2 = 2048.  Thus, 204.8 = 1024 / 5.

Do you know any other interesting math tricks?  Please share the trick below in the comments.

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10 Comments »

  1. rick December 4, 2012 at 7:54 am - Reply

    Nobody? Really? Bummer!

  2. Beksy December 14, 2012 at 5:05 am - Reply

    PiZZA was awesome! :) So were the others, thx really!

  3. Heather December 16, 2012 at 4:36 pm - Reply

    since when are 4 and 6 odd numbers?

    • Daven Hiskey
      Daven Hiskey December 16, 2012 at 6:50 pm - Reply

      @Heather: The “add up the odd number digits” refers to the odd numbered spaces within the number, so for 10604, that would be at position 1, position 3, and position 5: 1, 6, 4, in this case.

      • Heather December 21, 2012 at 9:20 pm - Reply

        Oh okay I understand now thank you that makes a lot more sense.

  4. t-bird December 21, 2012 at 7:18 pm - Reply

    Easy one: divisible by 5? Ends in 0 or 5
    Divisible by 9? Sum of the digits is divisible by 9

  5. t-bird December 21, 2012 at 8:11 pm - Reply

    …by the way, the 9′s trick means you can just keep adding and adding. If your final answer is 9, it was divisible by 9.

  6. David Gillies December 21, 2012 at 8:35 pm - Reply

    Because b – 1 | b^n – 1, the difference between any base b number and the number formed by any permutation of its digits is divisible by b – 1. Thus any decimal number minus a number formed by scrambling its digits is divisible by 9.

    Because the multiplicative order mod 7 of 10 is 6 (7 is a full reptend prime), (10^6 – 1)/9 = 142857 is a cyclic number. Successive multiples are cyclic permutations. 2, 3 , 4, 5, 6 x 142857 = 285714, 428571, 571428, 714285, 857142

    The divisibility by 7 trick works because (n – 21 n mod 10) mod 7 = n mod 7
    All of these are derived from modular arithmetic.

  7. brum September 6, 2013 at 6:19 am - Reply

    4 is not an ODD digit.

    • Daven Hiskey
      Daven Hiskey September 6, 2013 at 10:47 am - Reply

      @brum: As I stated above: The “add up the odd number digits” refers to the odd numbered spaces within the number, so for 10604, that would be at position 1, position 3, and position 5: 1, 6, 4, in this case.

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