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Tests for divisibility

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Integers

Tests for divisibility

Is 87 654 divisible by 3? You could do the division ā€” or you could add up the digits and know in about two seconds.

146Ć· 2 āœ“ones digit 6 is even632Ć· 4 āœ“last two digits 32 Ć· 4 = 848Ć· 8 āœ“8 Ɨ 6 = 48

What 'divisible' means

30 is divisible by 2 because 30 Ć· 2 has no remainder. It is also divisible by 3 and by 5. 30 is NOT divisible by 4, because 30 Ć· 4 = 7 remainder 2 (written 7 r 2). A whole number is divisible by 2 exactly when 2 is a factor of it.

The tests

Using 87 654 as the example: ā€¢ by 2 ā€” the last digit is 0, 2, 4, 6 or 8. Last digit is 4, so YES. ā€¢ by 3 ā€” the digits add to a multiple of 3. 8+7+6+5+4 = 30 = 10 Ɨ 3, so YES. ā€¢ by 4 ā€” the last TWO digits divide by 4. 54 Ć· 4 = 13 r 2, so NO. ā€¢ by 5 ā€” the last digit is 0 or 5. Last digit is 4, so NO. ā€¢ by 6 ā€” divisible by both 2 AND 3. Both hold, so YES. ā€¢ by 8 ā€” the last THREE digits divide by 8. 654 Ć· 8 = 81 r 6, so NO. ā€¢ by 9 ā€” the digits add to a multiple of 9. 30 is not, so NO. ā€¢ by 10 ā€” the last digit is 0. It is 4, so NO.

The two harder tests

BY 7 ā€” remove the last digit, then subtract twice it from what is left. If the result divides by 7, so does the original. 87 654 ā†’ remove the 4 to leave 8765. Then 8765 āˆ’ 2 Ɨ 4 = 8757. And 8757 Ć· 7 = 1252 exactly, so 87 654 IS divisible by 7. BY 11 ā€” add the digits in the odd places and the digits in the even places, then subtract. If you get 0 or a multiple of 11, the number is divisible by 11. For 87 654: odd places 4 + 6 + 8 = 18, even places 5 + 7 = 12. 18 āˆ’ 12 = 6, which is not 0 or a multiple of 11 ā€” so NO.

Combining tests

You can build bigger tests out of small ones. A number is divisible by 6 when it is divisible by 2 AND by 3. A number is divisible by 66 when it is divisible by 6 AND by 11. That only works when the two numbers share no factors ā€” which is why 6 = 2 Ɨ 3 works, but you cannot test for 8 by checking 2 and 4.

Worked examples

Is 87 654 divisible by 3?

  1. Add the digits: 8 + 7 + 6 + 5 + 4 = 30.
  2. 30 is a multiple of 3.

Answer: Yes

Is 87 654 divisible by 4?

  1. Look at the last TWO digits: 54.
  2. 54 Ć· 4 = 13 r 2, so it does not divide.

Answer: No

Is 87 654 divisible by 6?

  1. 6 = 2 Ɨ 3, so test both.
  2. Last digit 4 is even, so divisible by 2.
  3. Digits total 30, so divisible by 3.

Answer: Yes ā€” it passes both tests

Is 87 654 divisible by 7?

  1. Remove the last digit to leave 8765.
  2. Subtract twice the removed digit: 8765 āˆ’ 8 = 8757.
  3. 8757 Ć· 7 = 1251, exactly.

Answer: Yes

Why you cannot test for 8 using 2 and 4

  1. 2 and 4 share the factor 2.
  2. The combining trick only works when the two share no factors.

Answer: Use the last-three-digits test for 8 instead

Is 30 divisible by 4?

  1. 30 Ć· 4 = 7 remainder 2.
  2. A remainder means it does not divide exactly.

Answer: No

Where you see this in real life

  • Splitting a bill of Ā£4 728 between 8 people: the last three digits are 728, and 728 Ć· 8 = 91, so it divides exactly.
  • Checking whether 1 234 567 can be shared between 3 teams: the digits add to 28, which is not a multiple of 3, so it cannot.

Key words

Divisible
divides exactly, with no remainder
Remainder
what is left over after dividing
Tests for divisibility
quick checks for whether one number divides another
Factor
a number that divides into another exactly
Multiple
the result of multiplying a number by a whole number
Digit sum
the total you get by adding all the digits
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