Here we study methods that can be used to determine whether a number is evenly divisible by other numbers.

These are shortcuts for testing a number's factors without resorting to division calculations.

The rules given below transform a given number's divisibility by a divisor to a smaller number's divisibility by the same divisor.

If the result is not obvious after applying it once, the divisibility rule should be applied again to the smaller number.

We present the rules with examples, in a simple way, to follow, understand and apply.

We provide the divisibility rule for all the single digit divisors and prime divisors up to 47 and above and a general Divisibility Rule which is simple to use.

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Let 'd' ( > 10 ) be the divisor and 'D' ( > 10 ) be the dividend. We want to check the divisibilty of D by d.

Let D_{1} be the units' digit and D_{2} be the rest of the number of D. i.e. D = D_{1} + 10D_{2}

Let d_{1} be the units' digit and d_{2} be the rest of the number of d. i.e. d = d_{1} + 10d_{2}

To find out, whether D is divisible by d, find (D_{2}d_{1} - D_{1}d_{2})If you get an answer divisible by d (including zero), then D is divisible by d. If you don't know the new number's divisibility, you can apply the rule again.

Write D_{2}, D_{1}, d_{2}, d_{1} as follows.

D_{2} D_{1}

d_{2} d_{1}

Cross multiply and take the difference. If this difference is divisible by d, then D is divisible by d.

If you don't know the new number's divisibility, you can apply the rule again.

Let us see an example.

Find whether 8827 is divisible by 91.

Solution :

882 7

9 1

(882)(1) - (7)(9) = 882 - 63 = 819

81 9

9 1

(81)(1) - (9)(9) = 81 - 81 = 0

∴ 8827 is divisible by 91. Ans.

This divisibility rule is more convenient if the divisor 's last digit is small.

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To find out, if a number is divisible by 'p', take the last digit, multiply it by 'n', and add it to the rest of the number. or multiply it by '(p - n)' and subtract it from the rest of the number. If you get an answer divisible by 'p' (including zero), then the original number is divisible by 'p'. If you don't know the new number's divisibility, you can apply the rule again.

Finding 'n' for different prime numbers :

'n' is such that 'p' is the least factor of (10n - 1) or n = (kp + 1)/10 where k is the least natural number for which (kp + 1)/10 is a natural number (= n) or (kp + 1) is a multiple of 10.

For any prime number 'p', check for what value of k( = 1, 2, 3, 4, .......), (kp + 1) is a multiple of 10.

For Example

For 'p' = 13, (3)13 + 1 = 40 = a multiple of 10. So k = 3. ∴ n = (kp + 1)⁄10 = 40⁄10 = 4

For 'p' = 19, (1)19 + 1 = 20 = a multiple of 10. So k = 1. ∴ n = (kp + 1)⁄10 = 20⁄10 = 2

For 'p' = 29, (1)29 + 1 = 30 = a multiple of 10. So k = 1. ∴ n = (kp + 1)⁄10 = 30⁄10 = 3

The values of 'n' for prime numbers below 50 are

For 13, n = 4 and (p - n) = 13 - 4 = 9 For 17, n = 12 and (p - n) = 17 - 12 = 5 For 19, n = 2 and (p - n) = 19 - 2 = 17 For 23, n = 7 and (p - n) = 23 - 7 = 16 For 29, n = 3 and (p - n) = 29 - 3 = 26

For 31, n = 28 and (p - n) = 31 - 28 = 3 For 37, n = 26 and (p - n) = 37 - 26 = 11 For 41, n = 37 and (p - n) = 41 - 37 = 4 For 43, n = 13 and (p - n) = 43 - 13 = 30 For 47, n = 33 and (p - n) = 47 - 33 = 14

Note that for multiplying the last digit, for some prime numbers 'n' is more convenient and for some others '(p - n)' is more convenient.

If you select 'n', add ('n' x last digit), to the rest of the number. If you select '(p - n)', subtract {(p - n) x last digit}, from the rest of the number.

For example, for 19, n = 2 is more covenient than (p - n) = 17.

So, the divisibilty rule for 19 is as follows.

To find out, whether a number is divisible by 19, take the last digit, multiply it by 2, and add it to the rest of the number. If you get an answer divisible by 19 (including zero), then the original number is divisible by 19. If you don't know the new number's divisibility, you can apply the rule again.

For 31, (p - n) = 3 is more covenient than n = 28.

So, the divisibilty rule for 31 is as follows.

To find out, whether a number is divisible by 31, take the last digit, multiply it by 3, and subtract it from the rest of the number. If you get an answer divisible by 31 (including zero), then the original number is divisible by 31. If you don't know the new number's divisibility, you can apply the divisibility rule again.

Look at the divisibility rule for 7 above. There, we chose, '(p - n)' = 2 for 7. We can also choose 'n = 5' for 7 in which case we have to multiply the last digit by 5 and add it to the rest of the number to check its divisibilty by 7.

The method of finding 'n' given above can be extended to prime numbers above 50 also and the Divisibility Rule can be written for that.

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