Checking divisibility of a number

Standard

The ability to check whether a number is divisible by, say 4 or 9, seems to be useless at first.

But it could be a time saver when you’re trying to check the result of your calculation without resorting to the help of a calculator. Or it could be an amusing mental trick (or what Arthur Benjamin called Mathemagic) which you can use it for fun while drinking some beers in the pub.

Here are some of them:

– Divisibility by 2:

Very simple, check if the last digit is even or not. Thus 928474712 is divisible by 2, while 361635 is not.

– Divisibility by 3:

Quite simple, sum up all the digits in that number, and check if it’s multiples of 3.

Say you have a 7 digit number 9282648.

Sum up all the digits in that number, i.e: 9 + 2 + 8 + 2 + 6 + 4 + 8 = 39

39 is divisible by 3 (39 / 3 = 13), thus 9282648 is divisible by 3 ( 9282648 / 3 = 3094216 )

or you can go several steps further, keep summing up the digits, i.e. 3 + 9 = 12 then 1 + 2 = 3

– Divisibility by 4:

Check if the last 2 digits are divisible by 4( or if you didn’t memorize multiplication table of 4, then check if you can divide the last 2 digits of the number twice, since, you know 4 = 2 x 2 ).

Example: 948137924 is divisible by 4, since 24 is 6 x 4. ( 948137924 / 4 = 237034481 )

– Divisibility by 5:

Check if the last digit is 5 or 0 (e.g. 242452525, or 80808083130 )

– Divisibility by 6:

Similar to divisibility by 3, but the original number should be even number.

E.g. 9282648 is divisible by 6 and 3, while 9282645 is divisible by 3 but not by 6.

– Divisibility by 7:

For divisibility by 7, I only know the trick that uses recursive method.

  1. Separate the digits of the number into last digit and the rest of the digits.
  2. Do this: (rest of the digits) – (2 x last digit)
  3. Check if the result is multiple of 7, if yes, then voila, else repeat from step 1

Example: 2443

  1. last digit = 3, rest of the digits = 244
  2. 244 – (2 * 3) = 238
  3. Not sure if it’s divisible by 7
  4. last digit = 8, rest of the digits = 23
  5. 23 – ( 2 * 8 ) = 7
  6. Divisible by 7

– Divisibility by 8:

The last 3 digits should be divisible by 8. If it’s hard to check if 3 digits number is divisible by 8, try to divide it by 2 for 3 times (since 8 = 2 x 2 x 2)

Example: 24752, last 3 digits = 752.

If it’s hard to see whether 752 is divisible by 8, then keep subtracting 752 with 200, until you reached a number below 200, which is in this case 152, which is divisible by 8 (152 / 8 = 19).

Thus 24752 is divisible by 8 (24752 / 8 = 3094)

– Divisibility by 9:

Similar to divisibility by 3, but this time it will end up with multiples of 9, or if you keep doing it, it will end up with 9.

The sum of the digits of the result will always be 9.

Example: 1111111101 is divisible by 9, since 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 0 + 1 = 9. (1111111101 / 9 = 123456789)

or 7992 is divisible by 9, since 7 + 9 + 9 + 2 = 27, and 2 + 7 = 9. (7992 / 9 = 888)

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