Rules for Test of Divisibility

Test of divisibility is the most shorthand and useful way of checking whether a number is factor of the other, i.e., it helps to know if a number is a perfect divior of another number without actually dividing one by the other. Perfect divisors are those divisors which on dividing the dividends leave no remainders, i.e., reaminders in such cases are 0.

Divisibility by 1

Each and every number is divisible by 1. When we divide any number by 1 we always get the same number as result irrespective of the size of the number.

Divisibility by 2

A number is divisible by 2 if the digits in the unit place (ones place) is either of 0, 2, 4, 6, 8. That is, all the even numbers are divisible by 2.
Examples, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 etc.

Divisibility by 3

A number is divisible by 3 if the sum of all the digits in the number is a multiple of 3.
Examples, (i) 36 is divisible by 3, because 3+6=9 and 9 is a multiple of 3.
(ii) 726 is divisible by 3, because 7+2+6=15 and 15 is a multiple of 3.

Divisibility by 4

A number is divisible by 4 if the number formed by its last two digits are ether zero or a multiple of 4, then the original number is divisible by 4.
Examples,
(i) 128 is divisible by 4, because the number formed by the last two digits here is 28 and 28 is a multiple of 4.
(ii) 512 is divisible by 4 because the number formed with the last two digits here is 12 and 12 is a multiple of 4.

Divisibility by 5

A number is divisible by 5 if the number has either 0 or 5 in its ones place.
Examples,
(i) 15 is divisible by 5.
(ii) 700 is divisible by 5.

Divisibility by 6

A number is divisible by 6 if it is divisible by both 2 and 3.
Examples, 24, 30, 36, 414 all are divisible by 6 as they're already divisible by both 2 and 3.

Divisibility by 7

Step 1

A number is divisible by 7 if subtracting twice the last digit of the given number from the remaining digits gives a multiple of 7.
Examples,
91 is divisible by 7 because,
9-(2×1)=9-2= 7; and 7 is a multiple of 7.

Step 2

For bigger number like 39536, we make parts of the number, each part consisting of 3 digits starting from RHS.
In parts, we get 39 and 536
Now we find the difference between these numbers, if the difference is a multiple of 7, then the original number (i.e., 39536) is divisible by 7.
Here, 536-39= 497; and 497 can be found be a multiple of 7 (using Step1).
Therefore 39536 is divisible by 7.

Step 3

But what about a number which is bigger than the previous one?
Let us take a number 1025227
Making it in parts from RHS of 3-digits each we get 1, 025, 227.
There are 3 parts,
Let's add the first and last one,
1+227= 228
Now subtracting the left behind number from the sum 228, we get
228-025= 203; which is a multiple of 7, so the big number 1025227 is divisible by 7.

Let us take another number 3728823
3,728,823
3+823= 826
826-728 = 98; which is a multiple of 7
3728823 is divisible by 7.

Divisibility by 8

A number is divisible by 8 if the number formed by its last three digits are either 0 or a multiple of 8 then the original number is divisible by 8.
Examples, 43904, 413200, 850000 etc. are divisible by 8

Divisibility by 9

A number is divisible by 9 if the sum of all the digits of the number is multiple of 9.
Examples, 81, 405, 5004, 10098 etc. are divisible by 9

Divisibility by 10

A number is divisible by 10 if it has zero in its unit place.
Examples, 10, 20, 300 etc. are divisible by 10.

Divisibility by 11

A number is divisible by 11 if the difference of the sum of all the odd digits and the sum of all the event digits is zero or a multiple of 11.
Examples:
(i) 1331
Sum of all odd digits = 1+3= 4
Sum of all even digits = 3+1 = 4
Therefore difference = 4-4=0
Therefore 1331 is divisible by 11.

(ii) 120758
Sum of all odd digits = 8+7+2= 17
Sum of all even digits = 5+0+1 = 6
Therefore difference = 17-6=11
Therefore 120758 is divisible by 11.

(iii) 19487171
Sum of all odd digits = 1+1+8+9= 19
Sum of all even digits = 7+7+4+1= 19
Therefore difference = 19-19=0
Therefore 19487171 is divisible by 11.