Divisibility Rules

A divisibility rule is a way of determining whether a given number is divisible by a fixed number without performing the division.

We can use tests of divisibility to finding out the prime factors.


Tests for divisibility


Tests for divisibility by 2


If a number has any of the digits 0, 2, 4, 6, 8 in its units place, then that number is divisible by 2.

Simple Example


let’s say the number 3456
The number 3456 has 6 in its unit place. So that 3456 number is divisible by 2.


Medium Example:


Mark saved Rs.1200 in this month. He wanted to buy two gifts for his brother and sister. He thought to buy these gifts worth same cost. How would he spend his savings on these two gifts?
 He could spend Rs.600 per gift.

Explanation :

Mark had Rs.1200. He wanted to buy two gifts from this savings.
As he divided Rs.1200 in two equal parts.
The number 1200 has 0 in its unit place. So that1200 is divisible by 2.
Now 1200/2= 600.


Tests for divisibility by 3


If the sum of digits of number is divisible by 3 then the number is divisible by 3.

Simple Example


let’s say the number 3456.
The sum of the digits in the number 3456 is 3+4+5+6= 18. 18 is divisible by 3.
Hence, 3456 is divisible by 3.


Advance Example:


Jennifer decided to do the activity for Maths in the classroom. Classroom had total 78 students. She wanted to make 3 groups from the class with equal number of students. What would she do?
 Every group could have 26 students.

Explanation :

Classroom had total 78 students.
Check the number 78 with test for divisibility by 3.
The sum of the digits in the number 78 is 7+8=15.
15 is divisible by 3.
Hence, 78 is divisible by 3.
Now, 78/3= 26


Tests for divisibility by 4


If the number formed by digits in the tens and units places is divisible by 4, then that number is also divisible by 4.

Simple Example


let’s say the number 3112
In the number 3112, the number formed by the digits in the tens and unit places is 12.
The number 12 is divisible by 4 so that, the number 3112 is divisible by 4.


Advance Example


Rob has bundle of pages 5110. Would he able to distribute them with 4 clients equally?
 He would not able to distribute 5110 pages with 4 clients equally.

Explanation :

In the number 5110, the number formed by the digits in the tens and unit places is 10.
The number 10 is not divisible by 4 so that, the number 5110 is not divisible by 4.


Tests for divisibility by 5


If a number has either 0 or 5 in its units place, then that number is divisible by 5.

Simple Example


let’s say the number 3035.
The number 3035 has 5 in its unit place. So that 3035 is divisible by 5.


Tests for divisibility by 6


If a number can be divided by the numbers 2 as well as 3, then that number is divisible by 6.

Simple Example


1. let’s say the number 55128
The number 55128 has 8 in the unit place. There it is divisible by 2. The sum of the digits in the number 55128 is 5+ 5+1+2+8= 21 . 21 is divisible by 3. Therefore, 55128 is divisible by 6.


2. let’s say the number 45120.
The number 45120 has 0 in the unit place. There it is divisible by 2. The sum of the digits in the number 45120 is 4+5+1+3+0= 13. 13 is not divisible by 3. Therefore, 45120 is not divisible by 6.


Tests for divisibility by 9


If the sum of the digits in a number is divisible by 9, then that number is divisible by 9.

Simple Example


let’s say the number 55008.
The sum of the digits in the number 55008 is 5+5+0+0+8= 18. 18 is divisible by 9. Therefore, 55008 is divisible by 9.


Medium Example:


let’s say the number 2247.
The sum of the digits in the number 2247 is 2+2+4+7= 15. 15 is not divisible by 9. Therefore, 2247 is not divisible by 9.


Tests for divisibility by 10


If a number has 0 in its units place, then that number is divisible by 10.

Simple Example


let’s say the number 3050.
The number 3050 has 0 in its unit place. So that 3050 is divisible by 10.


Tests for divisibility by 11


If the difference between the sums obtained by adding alternate digits of the number is 0 or is divisible by 11 then that number is also divisible by 11.

Simple Example


let’s say the number 1463.
The sums of the alternate digits of the number 1463 are 1+6= 7 and 4+3= 7. the difference between them is 7-7= 0. Therefore, 1463 is also divisible by 11.


Medium Example:


let’s say the number 8243
The sums of the alternate digits of the number 8243 are 8+4= 12 and 2+3= 5. The difference between them is 12-5= 7. 7 is not divisible by 11.Therefore, 8243 is not divisible by 11.


Divisors of number.


The quotients obtained by dividing the number are the Divisors of number.
 To find divisor of 15, divide 15 by 3.
 We get 5 as quotients.
 So that, 3 and 5 is the divisor of 15.

Note:

  •  Every number is divisible by number 1 and itself.

Simple Example


 Divisors of 225 are 1, 3, 5, 9, 25, 45, 75, 225

Explanation:

225 is divisible by 5. 225 / 5 = 45.
225 is divisible by 3. 225 / 3 = 75
225 is divisible by 9. 225 / 9 = 25
Now every number is divisible by number 1 and itself.


Medium Example:


 Divisors of 420 are 1, 2, 3, 5, 10, 42, 70, 84, 140, 210, 420

Explanation:

We will use tests of divisibility here.
420 is divisible by 2. 420/2 = 210.
420 is divisible by 10. 420/10 = 42.
420 is divisible by 5. 420 /5 = 84.
420 is divisible by 3. 420 /3 = 140
420 is divisible by 2 and 3. So that, 420 is divisible by 6. 420 /6 = 70.
Now every number is divisible by number 1 and itself


Advance Example:


 Divisors of 8965 are 1, 5, 11, 815, 1993,8965

Explanation:

We will use tests of divisibility here.
You can see that 8965 has 5 in its unit place. Hence, 8965 is divisible by 5. 8965/5 = 1993.
The sums of the alternate digits of the number 8965 are 8+6= 14 and 9+5= 14.
The difference between them is 14-14= 0. Therefore, 8965 is also divisible by 11. 8965/11= 815
8965 is divisible by number 1 and itself.







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