Dear Job Aspirants,

In this page of number system we are going to
discuss about the Unit place determination, Number Divisibility by any divisor
and Number Progression. Number Divisibility simply means if any number is
divisible by any other Number or Not. Say 44 is divisible by 12 or not and like
so. In this tutorial we are going to focus on the Tips and tricks of Checking
Divisibility of any integer Number by other integer Number or not. We will also
focus on the Progression of Number system. We will mainly discuss the types of
progression those are Arithmetic Progression, Geometric Progression and
Harmonic Progression. Let’s go in Depth of Unit Place determination, Number
Divisibility and Progression by Topic Wise.

**Unit Place Determination:**To find out the Unit place of any product form Numeric or Index formed Numeric we should have some basic knowledge how the two digit and three digit Numbers are formed. Let have a look on this.

We all know two digit Numbers have Unit place
and ten’s place. And Three digit Numbers has Unit, Ten and Hundred’s places.
Let X, Y, Z be the three single valued integer Variable (Integer Variable means
it can take any integer value and single valued means it can take any value
from 0,1,2,3,4,5,6,7,8,9 these integers only). Then the any two digit Number
can be written as 10Y + X. e.g. 78 can be written as 10 *7 + 8 (here Y =7 and
X=8) similarly any three digit Number can be written as 100Z + 10Y +X. e.g. 128
can be written as 100*1+10*2+8 ( Here Z=1, Y=2, X=8).

*Tips & Trick 1:**Any Two digit Number (except pair Number i.e. 11 or 22 or so on) and the Number obtained by interchanging its Unit and Ten’s value is always a Multiplier of 9. E.g. 46 and 64 has a difference of 18 (64 – 46 =18 and 18 is multiplier of 9 i.e. 18 = 9x2). Similarly 78 and 87 has a difference of 9 (87-78 = 9, and 9=9x1) similarly 63 and 36 has a difference of 27 and 27 = 9 x 3.*

**How to Find Unit place Digit of any Product of Number**: If a product of Number is given to you and you are asked to find out the unit place Digit of this Product. Our general tendency to solve this kind of problem is that we try the conventional Multiplication technique and found the Unit place Digit of the product. But this technique is very time consuming. And in any Competitive examination time is very precious. So here I will give you technique to find out the Unit place of any product of Numbers. Let’s Discuss with an example.

Q.1 Find out the unit
Place digit of 2578 x 236 x 753.

To find Unit Digit first
take all the unit digit of Individual Numbers and Place as follow.

(..8)X(..6)X(..3) Now
multiply first two digit and write again as (..48)X(..3). Now again take the
unit digit of (..48) And write again as (..8)X(..3). Now get product value of 8
and 3 and found the Unit value here
8X3=24 so the Unit value is 4. So 4 is the Unit value of 2578 x 236 x
753.

**Now How to find out Unit placed Digit of any Number of Power Form:**Suppose you have been given a Number with a power value e.g. (357)^27 then how to determine the unit place digit. Since this is a very huge Number and Multiplying 357, 27 times is not practically possible in very short time. And thus finding Unit place digit is difficult in Conventional multiplication technique. So we must have to use a different technique to find out the Unit place digit. Some Useful step to remind is given below.

1. If
there is 0, 1, 5 and 6 in Unit place of original Number then whatever power is
given the unit place digit value will be the same. e.g. (1220)^287 will have a
unit place digit 0. Or e.g. (1156)^4728 will have Unit Place digit value of 6
always.

2. If
unit Place digit of given number is 9 then after the power value obtained the
unit place digit will be 1 or 9. It will be dependent on power value. If power
value is even Number the unit place value will be 1 and if Power value is Odd
Number then the Unit Place value will be 9. E.g. (49)^7 will have a unit place
digit 9 since power is odd. Again (49)^84 will have a unit place value of 1.

3. If
Unit place digit is 2 then the result unit place digit will be dependent on the
value of power. If power value is odd the unit place digit will be either 2 or
8. And if power value is even then unit place digit will be 4 or 6. e.g. (32)^7
here unit place value will be 8.

4. If
the unit place digit is 3 then we have to check the expression for 3^4. Because
3^4=1 so, any power of 1 will be 1. E.g.

(5443)^834 = (5443)^832 x (5443)^2

= ((…3)^4)^208 x (….3)^2

= (….1)^208 x (…9)

= (….1) x (…9)

= 9

5. If
the Unit place digit is 7 then also we have to check the expression for 7^4.
Because 7^4=1. So any power of 1 will be 1 and remaining calculation gives the
unit place digit. E.g.

(5447)^834 = (5447)^832 x (5447)^2

= ((…7)^4)^208 x (….7)^2

= (….1)^208 x (…14)

= (….1) x (…4)

= 4

6.
If Unit place digit is 4 then if power is
Odd Number then the final unit place digit will be 4 and if power is Even
Number then the unit place digit will be 6. E.g. (374)^577 here power of 374 is
577 which is odd Number so the unit place digit will be 4.

7.
If Unit place digit is 8 then fragment the
base Number in factor of 2 and find out the Unit place digit applying 6 No.
rule and 3 No. Rule. E.g.

(848)^104= (424x2)^104 =
(424)^104 X (2)^104

= (…6) X
((2)^2)^54

= (…6) X
(4)^54

= (…6) X (….6)

=36

So here the unit place
digit is 6.

Divisibility: Before
finding out the disability of any Number we should have to know Dividend, and
Divisor. The Number which is going to be divided is called Dividend, the Number
by which dividend is divided is called divisor. If Dividend is called D and
Divisor is called d the division operation can be represented as D/d. where “/”
is the Division operator. The result value of any division operation is called
Quotient. Quotient is represented as Q. Thus we can write any Number in terms
of Dividend, Divisor and Quotient if the Number is perfect multiple of
Quotient. Thus

**Dividend = Divisor X Quotient or D = d x Q or D/d = Q, Here D is perfect Multiple of d and Multiplying factor is Q.**

If the Dividend Number D
is not Perfect multiple of Divisor d the D is not perfectly divided by d and
some value remain which is called Remainder R. Thus in such Numbers are
represented as:

**Dividend = (Divisor X Quotient) + Remainder**

**D = (d x Q) + R**

**e.g.78 = (12x6)+6, where 78 is dividend, 12 divisor,6 Quotient and 6 is remainder**

**Divisibility Testing of Any Number:**If a Number is given to you and you are asked to find the check whether the Number is divisible by any divisor r then we generally try to proceed to conventional Division procedure. But this long division procedure is very much time consuming. Thus we must have some short cut techniques to check the divisibility of any Number by given divisor. Some short cut technique to check divisibility is given below:

1.

**Divisible by 2**: If unit Place digit is even Number or Zero. e.g. 48,72,8678, 50,1000
2.

**Divisible by 3**: if the summation of all digits of given Number is divisible by 3 then the number is also divisible by 3 e.g 7824 is divisible by 3 because summation of all digits are 7+8+2+4= 21 thus 21 is divisible by 3 i.e. 21=7x3.
3.

**Divisible by 4**: if last two digit of any Number is divisible by 4. E.g. 4276 here last two digit 76 is divisible by 4 i.e. 76=19X4. So the Number 4276 is divisible of 4.
4.

**Divisible by 5**: If the last digit of any Number is 0 or 5 then the Number is divisible by 5.
5.

**Divisible by 6**: if any number is divisible by 2 and also divisible by 3 then the Number is divisible by 6 also. E.g. 726 is divisible by 2 and 3 both. So 726 is divisible by 6.
6.

**Divisible by 7**: if the difference between twice the value of Unit place digit and the Summation of other remaining digits is 0 or multiple of 7 then the Number is divisible by 7. E.g. 728 here Unit place digit is 8 and twice of it is 8x2=16 and summation of other two digit is 7+2=9 so difference between 9 and 16 is 7. So the Number is divisible by 7. Keep in mind here we are talking about difference value. So it will be positive. Always minus the smaller one from bigger one.**7.**

**Divisible By 8:**If last three or more digits of any Number is 0 or if the Number formed by last three digit is divisible by 8 then the number is also divisible by 8. E.g. 789000 has last three digit zeros, so Number is divisible by 8. Again 87624 has last three digits 624 which is divisible by 8. So 78624 is also divisible by 8.

8.

**Divisible by 9:**If the summation of all digits of given Number is divisible by 9 then the number is also divisible by 9 e.g. 78246 is divisible by 9 because summation of all digits are 7+8+2+4+9= 27 thus 27 is divisible by 9 i.e. 21=9x3.**9.**

**Divisible By 10:**If any Number given has Unit place digit of 0 then the Number is always divisible by 10. E.g 7680, 1000,220, 560, 7682000, 380 etc.

**10.**

**Divisible by 11:**If the difference value of Sum of Even place digits and sum of Odd placed digits is 0 or multiple of 11 then the Number is divisible of 11. E.g. 105906944 has Odd placed digits 4,9,0,5,1 and summation is 4+9+0+5+1=19 and even placed digits 4,6,9,0 and summation is 4+6+9+0=19 and the difference is 19-19=0.Thus the Number is divisible of 11.

**11.**

**Divisible by 12:**If the given Number is divisible by 3 and 4 both then the Number is divisible of 12 also. E.g. 27216 is divisible of 3 as well as 4. So the number is also divisible by 12.

**12.**

**Divisible of 14:**if the Number is divisible by 7 and 2 both then the Number is divisible of 14 also. E.g.686 is divisible by both 2 and 7 thus divisible by 14 also.

**13.**

**Divisible of 15:**If the given Number is divisible by 3 and 5 Both then the Number will be divisible by 15 also.

**In similar way we can found the divisibility by 18,21,24,27 etc.**

**Divisibility testing for repetitive digit Numbers:**

**Any repetitive 6 digit Number is always divisible by 7,11 and 13 e.g. 777777 or 999999 is divisible by 7,11 and 13.**

Any number repeating in
two digits three times is always divisible by 7 e.g. 323232, 474747 etc.

Any number repeating in
three digits two times is always divisible by 7,11 and 13 e.g. 326326, 476476
etc.

**Number System Progression:**Progression of Number system is how a series of Numbers follow certain pattern. E.g. 1,3,5,7,…. In this series of number a certain pattern is followed i.e. each number is differ by two positions. This kind of series is called Number system Progression. There are mainly three types of Number system progressions.

**1.**

**Arithmetic Progression 2. Geometric Progression 3. Harmonic Progression**

**Arithmetic Progression:**In Arithmetic Progression there is always a difference exist between two consecutive Numbers. e.g.1, 4, 7, 10, 13, 16….. N where N is any integer Number. In arithmetic’s progression the Summation of N number is obtain with the help of following Equation:

**Sn=n/2[2a + (n-1)d]**

Where,

Sn=
Summation value of the series.

n=
number of terms in the series

a=
first term of the series

d=
difference between two consecutive term

e.g.
1,2,3,4,……….84, here total Number of terms = 84 so n=84, d= 2-1=1, a=1

Here summation of nth term will be Sn=84/2[2.1
+ (84-1).1]= 42[2+83]=42*85=3570

Nth
term of any arithmetic series can be obtained with the following formula:

**Tn= a +(n-1)d**.

*If a, b, c are in Arithematics progression then*

*b = (a+c)/2***Geometric Progression:**In geometric progression of any number series there exist a common ratio between two consecutive numbers. e.g. 2,4,8,16,32…. Here ratio of 4/2 =2 and 8/4=2 and so on. So here a common ratio 2 is existing for all consecutive Numbers. This type of series is called Geometric Series or Geometric Progression of Numbers.

The
sum of nth terms of Geometric Series can be found with the Help of following
Expression:

**Sn=a(1- r^n)/(1- r), if r<1 and……..(i)**

**Sn= a(r^n - 1)/(r - 1); if r>1 ………..(ii)**

Where
r = common ratio,

a = first term

Let
we have geometric series as 1,4,16,64,216…..10terms here common ratio is
4/1=16/4=4 and first term is a= 1 so sum of ten terms will be as follow:

Since
here r>1 so equation (ii) will be followed. So

Sn=1(4^10-1)/
(4-1) =(4^10-1)/3 = 1048575/3 = 349525.

The
nth tern of Geometric series is found with the help of following equation:

**Tn= ar^(n-1)**

*If the Series is infinite or never ending then the Summation will be as follows*

*Sinf=a/(1-r)*

*If a, b, c are in geometric progression then*

*b =sqrt(ac)***Harmonic Progression:**If we inverse the each term of any series and if there exist a common difference between two consecutive Numbers then the progression is called Harmonic Progression. E.g.

**1/3, 1/6, 1/9, 1/12**

*If a, b, c are in Hermonic progression then*

*b = 2ac/(a+c)*
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