# Divisibility Rules /Test

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 byʿ9ʾ :
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 numberThe 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.     Whole number: 0 (Zero) and set of natural numbers are all together called the 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.      Even numbersAll positive  numbers which are divisible by 2 are called
Even number.
So, Even number={ 0,2,4,6,8,10,12,14,………………∞ }
5.      Odd numbersThe 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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