Suppose that, you are asked to simplify a fraction say 126/216 . Since the numbers are relatively bigger, the task is not easy.

**Rules
for Test of Divisibility of Numbers**

Suppose that, you are asked to simplify a fraction
say 126/216 . Since the numbers are relatively bigger, the task is not easy. Observe
that, these numbers are not only divisible by 2 and 9 exactly but by other numbers
too! How do we know that 2 and 9 are factors of 126 and 216? We are going to see
** divisibility tests** in this section
which are rules that will improve your mental math skills for such determinations.

Divisibility tests, in common, are useful in the
prime factorisation of a number. Also, it is fun to find whether any large number
is exactly divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10 or 11 (and more...) by simply
observing, examining and performing basic operations with the digits of the given
number and not by doing the actual division Curious to know? Then, remember the
following interesting rules and have fun...! As divisibility by 2, 3 and 5 gain
importance in the prime factorisation of a number, we will define the rules for
them first!

__Divisibility by 2__

**A number is divisible
by 2, if its ones place is any one of the even numbers 0, 2, 4, 6 or 8.**

**Examples:**

1. 456368 is divisible
by 2, since its ones place is even(8).

2. 1234567 is not divisible
by 2, since its ones place is not even(7).

__Divisibility by 3__

Divisibility of a number by 3 is interesting! We
can find that 96 is divisible by 3. Here, note that the sum of its digits 9+6 =
15 is also divisible by 3. Even 1+5 = 6 is also divisible by 3. This is called as
*iterative* or *repeated* addition. So,

**A number is divisible
by 3 if the sum of its digits is divisible by 3.**

**Examples:**

1. 654321 is divisible
by 3.

Here 6+5+4+3+2+1= 21 and 2+1=3 is divisible by 3.

Hence, 654321 is divisible by 3.

2. The sum of any three
consecutive numbers is divisible by 3.

(For example: 33+34+35=102, is divisible by 3)

3. 107 is not divisible by 3 since 1+0+7=8, is not
divisible by 3.

__Divisibility by 5__

Observe the multiples of 5. They are 5, 10, 15,
20, 25,.., 95, 100, 105, …., and keeps on going.

It is clear, that multiples of 5 end either with
0 or 5 and so,

**A number is divisible
by 5 if its ones place is either 0 or 5.**

**Examples: **5225 and 280 are divisible by 5

**Try these**

**(i) Are the leap years divisible by
2?**

Yes. The leap years are
divisible by 2.

**(ii) Is the first 4 digit number divisible
by 3?**

The first 4 digit number 1000

It is not divisible by 3.

**(iii) Is your date of birth (DDMMYYYY)
divisible by 3?**

Date of Birth 09−06−2007.
It is divisible by 3.

**(iv) Check whether the sum of 5 consecutive
numbers is divisible by 5.**

The sum of 5 consecutive numbers is divisible 5

1 + 2 + 3 + 4 + 5 = 15/5
= 3

16 + 17 + 18 + 19 + 20 = 90/5 = 18

**(v) Identify the numbers in the sequence
2000, 2006, 2010, 2015, 2019, 2025 that are divisible by both 2 and 5.**

The numbers divisible by 2 are 2000, 2006, 2010

The numbers divisible by 5 are 2000, 2010, 2015

__Divisibility by 4__

**A number is divisible
by 4 if the last two digits of the given number is divisible by 4.**

Note that **if the last two digits of a number
are zeros, then also it is** **divisible by 4.**

**Examples: **71628, 492,
2900** **are divisible by 4, because** **28** **and** **92**
**are divisible by 4 and** **2900 is also
divisible by 4 as it has two zeros.

__Divisibility by 6__

**A number is divisible
by 6 if it is divisible by both 2 and 3.**

**Examples: **138, 3246, 6552 and 65784 are divisible by 6.

**Note**

Though a rule for divisibility of a number
by 7 exists, it is a bit tricky and dividing directly by 7 will be easier.

__Divisibility by 8__

**A number is divisible
by 8 if the last three digits of the given number is divisible by 8.**

Note that **if the last three digits of a number
are zeros, then also it is** **divisible by 8.**

**Examples: **2992** **is divisible by 8 as** **992** **is divisible by 8 and 3000** **is divisible by 8 as its** **last three
digits are zero.

__Divisibility by 9__

**A number is divisible
by 9 if the sum of its digits is divisible by 9. **Note that the** **numbers divisible by 9 are
divisible by 3.

**Examples: **9567 is divisible by 9 as 9+5+6+7=27 is divisible by 9.

__Divisibility by 10__

**A number is divisible
by 10 if its ones place is only zero. **Observe that numbers divisible** **by 10 are also divisible by 5.

**Examples:**

1. 2020 is divisible
by 10 (2020÷10 = 202) where as 2021 is not divisible by 10.

2. 26011950 is divisible
by 10 and hence divisible by 5.

__Divisibility by 11__

**A number is divisible
by 11 if the difference between the sum of alternative digits of the number
is either 0 or divisible by 11.**

**Examples: **Consider the number** **256795. Here, the difference between the sum of** **alternative
digits = (2+6+9 )−(5+7+5)=17−17=0.

Hence, 256795 is divisible by 11.

**Activity**

The teacher may ask all the students
to check mentally for divisibility by 2, 3, 4, 5, 6, 8, 9, 10 and 11. If divisible,
let them write ‘yes’, otherwise ‘no’ (the first one is done for you!).

Tags : Numbers | Term 2 Chapter 1 | 6th Maths , 6th Maths : Term 2 Unit 1 : Numbers

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6th Maths : Term 2 Unit 1 : Numbers : Rules for Test of Divisibility of Numbers | Numbers | Term 2 Chapter 1 | 6th Maths

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