# NUMBER SYSTEM CHAPTER 3

**TEST FOR**

**DIVISIBILITY**

__DIVISIBLE BY’2’:__
When
the last digit of a number is either 0 or even, then the number is divisible by
2.

Example: 26,58,20,46,1256,532656
…………….. etc.

__DIVISIBLE BY ‘3’:__
When the
sum of the digits of a number is divisible by 3, then the number is divisible
by 3.

Example:

1533 1+5+3+3=12, which is divisible by 3 so 1533
is also divisible by 3.

126 1+2+6=9, which is divisible by 3 so 126 is also
divisible by 3.

96369 9+6+3+6+9=33, which is divisible by 3 so 96369
is also divisible by
3

__DIVISIBLIE BY ‘4’:__
When the last two digits are divisible by 4
or zeros, it is multiple by 4.

Example:

6428 last two digits is 28 it is divisible by 4
so the example is also divisible by 4.

15200 last
two digits are 00 so its divisible by 4 the example also divisiblee by 4.

__DIVISIBLE BY ‘5’:__
Which number
having 0 or 5 at the end of the number its divisible by 5.

Example:

45,90,8520,6235,695425,60000……….. etc.

**:**

__DIVISIBLE BY’6’__
When the
number divisible by both 3and 2,then the perticular number is divisible by 6
also.

Example:

96,720,540,256,…………etc.

__DIVISIBLE BY’7’:__
A number is divisible by 7 when the
differnce between twice the digit at once place the number formed by other
digits is zero or multiple of 7.

Example:

658=
65-2*8 =49 here the unit place is 8 its multiple by 2 and the difference between
the number is 49. So the number is divisible by 7.

__DIVISIBLE BY’8’:__
When the
last three digits are divisible by 8 or zeros
the number is divisible by 8. Last three or more digits of a number are
zero is also divisible by 8.

Example:

4456
the last three digits are divisible by 8,so the number also divisible by 8.

__DIVISIBLE BY’9’:__
The sum
of the all digits of a number is divisible by 9,then the number is divisible by
9.

Example:

936819 9+3+6+8+1+9=36 which is divisible by 9 the
number also divisible by 9.

123456789
1+2+3+4+5+6+7+8+9=45 which is divisible by 9 number also divisible by 9.

__DIVISIBLE BY ‘10’:__
When the
last number ends with zero the number is divisible by 10.

Example:

250,63520,369852140,2500,36000,…………… etc.

__DIVISIBLE BY’11’:__
When the
sum of the digits at odd and even places are equal or differ by 11,then the
number also divisible by 11.

Example:

217382 sum
of digits in odd places:2+7+8=17

Sum of digits in even places:1+3+2=6

The difference is 11. So the number is divisible by 11.

__DIVISIBLE BY ‘12’:__
The number
which is divisible by 4 and 3 is also
divisible by 12.

Example:

2244
is divisible by both 3 and 4 so the number is divisible by 12.

3648 is divisible by both 3 and 4 so the number is divisible by 12.

__DIVISIBLE BY ‘125’:__
A number is divisible by 125 when the number made by last three digits
are divisible by 125.

Example:

5684125 is divisible by 125 as
the last three digits of the number is divisible by 125,so the number is
divisibnle by 125.

65845250 is divisible by 125 as
the last three digits of the number is divisible by 125,so the number is
divisibnle by 125.

**BASIC NUMBER THEORY**

·
Square of every even number is an even
number while squre of odd number is an odd number.

·
A number obtained by squaring a number does
nit have 2,3,5,7 or 8 at its unit place.

·
Sum of frist natural numbers =

__n(n+1)__
2

·
Sum of frist n odd numbers= n2

·
Sum of frist n even numbers= n(n+1)

·
Sum of square of frist n natural numbers =

__n(n+1)(2n+1)__
6

·
Sum of cubes of frist n natural numbers =[

__n(n+1)__]2
2

·
For any natural number n,(n3-n) is
divisible by 6.

·
The product of three consecutive number is
always divisible by 6.

__Arithmetic series:__
a,(a+d),(a+2d),(a+3d),………………………

a= 1

^{st}term, d= common differnce.
Nth term=
a+(n-1)d

Sum of n terms = n/2[2a+(n-1)d]

Sum of n terms = n/2 (a+d),
where i= last term

__Geometric series:__

a,ar,ar2,ar3,………

a= 1

^{st}term r= common ratio,then
sum of n
terms= a(1-rn)/(1-r) where r<1

sum of n
terms = a(1-rn)/(r-1) where r>1

NUMBER SYSTEM CHAPTER 3
Reviewed by
Honey
on
14:30:00
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