Divisibility Rules

Divisibility Rules:

The Process of checking whether a number is divisible by a given number or not without actual division is called divisibility rule for that number.
There are certain tests for divisibility of numbers by any of the numbers 2,3,4,5,6,8,9, 10 and 11 such that by simply examining the digits in the given number, one can easily determine whether or not a given number is divisible by any of these numbers.

Test of Divisibility by 10:

A number is divisible by 10, if its unit’s digit is zero.
Each of the numbers 70, 120, 1550, 60, 240,…etc., are divisible by 10.
None of the numbers 54, 26, 69,…etc., are divisible by 10.

Test of Divisibility by 5:

A number is divisible by 5, if its units digit is either O or 5.
Each of the numbers 55, 115, 210, 10575 is divisible by 5. However, none of the numbers 127, 89451, 1326 is divisible by 5.

Test of Divisibility by 2:

A number is divisible by 2, if its units digit is either O, 2, 4, 6 or 8.
Each of the numbers 22, 54, 72 is divisible by 2 but none of the numbers 727, 15423, 5871 is divisible by 2.

Test of Divisibility by 3:

A number is divisible by 3 if the sum of its digit is divisible by 3.
Consider the number 2418.
We have,
Sum of the digits = 2 + 4 + 1 + 8 = 15, which is divisible by 3.
So, 2418 is divisible by 3.
Consider the number 43249.
We have,
Sum of the digits = 4 + 3 + 2 + 4 + 9 = 22, which is not divisible by 3.
Hence, the number 43249 is not divisible by 3.

Test of Divisibility by 9:

A number is divisible by 9, if the sum of its digits is divisible by 9.
Consider the number 45063.
We have,
Sum of the digits = 4 + 5 + O + 6 + 3 = 18. which is divisible by 9.
Therefore, 45063 is divisible by 9.
Now, consider the number 5412.
For this number, we have.
Sum of the digits = 5 + 4 + 1 + 2 = 12. which is not divisible by 9.
So, 5412 is not divisible by 9.

Test of Divisibility by 4:

A number is divisible by 4, if the number is divisible by 2 twice (or) number formed by its digits in ten’s and unit’s places is divisible by 4.
Consider the number 78936.
The number formed by ten’s and unit’s digits is 36, which is divisible by 4. Therefore 78936 is divisible by 4.
Consider the number 90873.
For this number, the number formed by ten’s and unit’s digits is 73, which is not divisible by 4. Therefore, 90873 is not divisible by 4.

Test of Divisibility by 6:

A number is divisible b 6, if it is divisible by both 2 and 3.
Consider the number 75642.
Its unit’s digit is 2. So, it is divisible by 2. ‘
Sum of its digits 7 + 5 + 6 + 4 + 2 = 24, which is divisible by 3. Therefore, the given number is also divisible by 3. Thus, the given number is divisible by both 2 and 3. Hence, it is divisible by 6.
Consider the number 56427.
Its unit’s digit is 7, so it is not divisible by 2. Hence, it is not divisible by 6.
Consider the number 29356. Its unit’s digit’is 6. So, it is divisible by 2.
Sum of its digits = 2 + 9 + 3 + 5 + 6 = 25, which is not divisible by 3.
So, 29356 is not divisible by 3. Hence, 29356 is not divisible by 6.

Test of Divisibility by 8:

A number is divisible by 8, if the number formed by its digits in hundred’s, ten’s and unit’s places is divisible by 8.
Consider the number 569288.
The number formed by hundred’s, ten’s and unit’s digits of this number is 288, which is divisible by 8. Therefore, 569288 is divisible by 8.
Consider the number 965214.
The number formed by hundreds, ten’s and unit’s digits of this number is 214. which is not divisible by 8. Therefore, 965214 is not divisible by 8.

Test of Divisibility by 11:

A number is divisible by 11, if the difference of the sum of its digits in odd places and the sum of its digits in even places (starting from unit’s place) is either O or a multiple of 11.
Consider the number 61809.
Sum of its digits in odd places = 9 + 8 + 6 = 23
Sum of its digits in even places = O + 1 = 1
Difference of the two sums = 23 — 1 = 22, which is divisible by l1
Therefore, 61809 is divisible by 11.
Consider the number 8050314052
We have,
Sum of the digits in even places = 5 + 4 + 3 + 5 + 8 = 25
Sum of the digits in odd Places = 2 + O + 1 + O + O = 3
Difference of these sums = 25 —3 = 22, which is divisible by 11
Therefore, the given number is divisible by 11.

Test of Divisibility by 7:

A number is divisible by 7, if the difference between twice the unit’s digit and the number formed by other digits is either 0 or a multiple of 7.
Consider the number 6804.
We have,
(680 – (2 x 4)) = 672, which is divisible by 7.
Therefore, 6804 is divisible by 7.
Consider the number 137.
Clearly, (2 x 7) – 13 = 1, which is not divisible by 7.
Therefore, 137 is not divisible by 7.

Properties of Divisibility:

Property 1:
If a number is divisible by another number, then it is divisible by each of the factors of that number.
OR
If a, b, c are three natural numbers such that a is divisible by b and b is divisible by c, then a is divisible by c also.
Verification: We know that 72 is divisible by 6 and 6 is divisible by 2 and 3. By using tests of divisibility by 2 and 3, we find that 72 is also divisible by 2 and 3.
Consequences:
1) Since 9 is divisible by 3. Therefore, every number divisible by 9 is also divisible by 3.
2) Since 4 is divisible by 2. Therefore, every number divisible by 4 is also divisible by 2.
3) Since 6 is divisible by 2 and 3 both. Therefore, every number divisible by 6 is also divisible by both 2 and 3.

Property 2:
If a number is divisible by each of the two or more co-prime numbers, then it is divisible by their products.
OR
If a and b are two co-prime numbers such that a number c is divisible by both a and b, then c is also divisible by a x b.

Verification: Consider the number 67542. Its unit’s digit is 2. So, it is divisible by 2.
Sum of its digits = 6 + 7 + 5 + 4 + 2 = 24, which is divisible by 3. So, it is divisible by 3 also.
Since 2 and 3 are co-prime numbers. Therefore, 67542 must be divisible by 6 also. By actual division, we find that it is exactly divisible by 6.

Consequences:
1) Since two prime numbers are always co-prime. Therefore, if a number is divisible by each one of any two prime numbers, then the number is divisible by their product.
2) Since 2 and 3 are co-primes. Therefore, if a number is divisible by both 2 and 3, it must be divisible by 2 x 3 = 6.

Property 3:
If a number is a factor of each of the two given numbers, then it is a factor of their sum.
OR
If two numbers b and c are divisible by a number a, then b + c is also divisible by a.

Verification: We know that 75 and 125 both are divisible by 5. The sum of these two numbers is 75 + 125 = 200. By actual division, we find that the sum 200 is also divisible by 5.

Property 4:
If a number is a factor of each of the two given numbers, then it is a factor of their difference.
OR
If two numbers b and c are divisible by number a, then b — c is also divisible a.

Verification: We know that 381 and 465 are divisible by 3, because sums of their digits are divisible by 3.
Difference of these two numbers = 465— 381 = 84
Clearly, Sum of its digits = 8 + 4 = 12, which is divisible by 3.
So, 84 is divisible by 3.