**Divisibility Test/Rules by 0,1,2,3,4,5,6,7,8,9,10,11,12**

*1.*

__Divisibility by__

*ʿ0ʾ :***No number is divisible by 0. the result of divisibility by 0 to any number is infinite or indeterminate or unknown.**

Any number ʿxʾ

**divisible by 0 =****( infinite).**

*2.*

__Divisibility by__

*ʿ1ʾ :*

**If we**

**divide any**

**number by 1 then we always get that number.**

**3.**

__Divisibility by__**:**

*ʿ2ʾ***If unit digits of a number are 0,2,4,6,8 then that number is always divisible by 2.**

**Eg .**

**222, 1252, 72, 95855445548, 254556, 5111141200 etc.**

*4.*

__Divisibility by__

*ʿ3ʾ :***If the sum of all digits of a number is divisible by 3 then that number is divisible by 3.**

**Eg.**

**Suppose 3488247 have to be divided by 3, we see 3+4+8+8 +2+4+7=36 which is divisible by 3. So 3488247 is divisible by 3.**

**As same in 54655 see 5+4+5+5=19 which is not divisible by 3. So 54655 is not divisible by 3.**

*5.*

__Divisibility by__

*ʿ4ʾ :***A number is divisible by 4 if number form by its last two digits is divisible by 4.**

Eg.

**45750124 is divisible by 4 because of its last two digits is 24 and it is divisible by 4.****545808 is divisible by 4 because of its last two digits is 08 and it is divisible by 4.**

312546 is not divisible by 4 because of its last two digits is 46 and it is not divisible by 4.

*6.*

__Divisibility by__

*ʿ5ʾ :***A number having unit digit only 0or 5 is divisible by 5.**

**Eg.**

**15, 225, 1250 ………etc are divisible by 5 because of their unit digits have 0 or 5.**

**46,21,203 ……..etc are not divisible by 5 since their unit digit not containing 0 or 5.**

*7.*

__Divisibility by__

*ʿ6ʾ :***A number is divisible by 6 if it is divisible by both 2 and 3.**

**Eg**

**. 749562 is divisible by 6 because it is divisible by both 2 and 3.**

**3324 is divisible by 6 because it is divisible by both 2 and 3.**

**8.**

__Divisibility by__

*ʿ7ʾ :***A number is divisible by 7 if the difference between twice the last digits and the number formed by other remaining digits of the number is either 0 or multiple of 7or divisible by 7.**

**Eg. A)**

**Suppose 254 have to be divided by 7, here**

**Step1**

**: Last digit x2=4x2=8**

**Step2**

**: Number formed by remaining digits=25**

**Step3**

**: Different between them= 25-8=17, Which is neither 0 nor multiple of 7. So 254 is not divisible by 7.**

**B)**

**In the case of 7854**

**Step1**

**: Last digit x2=4x2=8**

**Step2**

**: Number formed by remaining digits=7854**

**Step3**

**: Different between them= 1785-8=777, Which is multiple of 7. So 7854 is divisible by 7**.

**9.**

__Divisibility by__

*ʿ8ʾ :***A number is divisible by 8 if the number formed by last three digits is divisible by 8.**

**Eg**

**. 25480 is divisible by 8 because its last three digits 480 is divisible by 8.**

**1257 is not divisible by 8 because its last three digits 257 is not divisible by8.**

**10.**

__Divisibility by__

*ʿ9ʾ :***A number is divisible by 9 if the sum of all of the number is divisible by 9.**

**Eg**

**. 556362 is divisible by 9 because the sum of its all digits is divisible by 9.**

**698555 is not divisible by 9 because the sum of its all digits is not divisible by**

**9.**

**11.**

__Divisibility by__

*ʿ10ʾ :***All numbers having its unit digit zero (0) are always divisible by 10.**

**Eg**

**. 10, 120, 70, 444444440 etc all are divisible by 10.**

**12.**

__Divisibility by__

*ʿ11ʾ :***A number is divisible by 11 if the difference between the number of the sum of its odd places digits and even places digits is 0 or multiple of 11or divisible by 11.**

**Eg**

**.230395 is divisible by 11 because**

**Sum of odd places digits,2+0+9=11**

**Again sum of even places digits, 3+3+5=11**

**And their different, 11-11=0, so 230395 is divisible by 11.**

**13.**

__Divisibility by__

*ʿ12ʾ :***A number is divisible by 12 if it is divisible by both 3 and 4.**

**Because 12 = 3 × 4.**

**Eg**

**. 331728 is divisible by 12 because the sum of all digits i.e. 3+3+1+7+2+8=24 is divisible by 3 and also last two digits 28 is also divisible by 4 and so 331728 is divisible by both 3 & 4.Hence 331728 is divisible by 12.**

**?**

*# What is Mathematics*
=) Mathematics is a bone branch of science and Technology which has developed by different Civilizations of the world.

__Concepts about numbers we have been using__

__–__*1.*

__Natural number__*:*The numbers which create naturally by ancient human civilization to

exchange, communication, commerce, bargain, relations etc are called natural numbers

*.*
Natural numbers are denoted by 'N'.

And N={1,2,3,4,5,6,7,8,9,10,11,…………………………………………∞}.

2.

*: 0 (Zero) and set of natural numbers are all together called the Whole number.*__Whole number__
The whole number is denoted by ‘W’

Every natural is a whole number.

W= { 0,1,2,3,4,5,6,7,8 ,9,10,11,12 ………………………………….α}

3. Prime Number: A number which has only two (2) factors that`s of one is itself and another is 1 is called a Prime number.

The smallest prime number is 2.

*Eg*

*.*2=1x2 or 2x1 ,so total factors 2 i.e. 1&2 so 2 is a Prime number.

3=1x3 or 3x1, so total factors 2 i.e. 1&3 so 3 is a Prime number.

11=1x11 or 11x1 ,so total factors 2 i.e. 1&11 so 11 is a Prime number.

23=1x23 or 23x1 ,so total factors 2 i.e. 1&23 so 23 is a Prime number

But 4=1x4 or 2x2, so there are total 3 factors i.e. 1,2,4 and so 4 is not a Prime number.

6=1x6 or 2x3, so there are 4 factors i.e. 1,2,3,6 and so 6 is not a Prime number.

4.

**: All positive numbers which are divisible by 2 are called**__Even numbers__**Even number.**

So, Even number={ 0,2,4,6,8,10,12,14,………………∞ }

5.

__Odd numbers__: The positive numbers which are not divisible by 2 are called Odd number.
So, Odd number ={ 1,3,5,7,9,11,13,15,17,19 …………………..∞ }.

6. Integer numbers: All natural numbers, negative of natural numbers and including zero are called Integer numbers.

Generally, the Integer number is denoted by ‘ I’ or ‘Z’.

And I = { 0 ±1, ±2,±3,±4,±5,±6,±7,±8,±9,±10,…………………∞ }.

The Integer numbers are divided into two parts as given below-

a)Positive Integer: All whole numbers are called positive (+ve ) Integer.

It is denoted by

__ʻ I___{+}ʼ and_{ }I_{+ }={ 0,1,2,4,5,6,7,…………………. ∞}
b) Negative Integer:

__All whole numbers containing negative sign in front of are call negative Integers.__
It is denoted by ‘I

_{-}ʼ and I_{-}={ 0,-1,-2, -3, -4, -5, -6, ………………..∞}**********

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