# Divisibility Rule of Numbers

Are you suffering to remembering Divisibility rule? Don’t worry, just follow this post and definitely you will remember all the rule.

## Divisibility Rule

### Divisibility by 2

All numbers are divisible by 2 if its first digit from your right side (Unit Place) contains 0, 2, 4, 6 or 8. Only all, even numbers are divisible by 2.

Number Unit Place Is divisible? Cause

110

0
Yes
Its Unit Place is 0
192
2
Yes
Its Unit Place is 2
234
4
Yes
Its Unit Place is 4
366
6
Yes
Its Unit place is 6
568
8
Yes
Its Unit Place is 8
651
1
No
Its Unit Place is 1
96767
7
NO
Its Unit Place is 7

### Divisibility by 3

A number is divisible by 3 only if the sum total of all the digits is divisible by 3. We have to add all the digits of that number and check whether the number is divisible by 3 or not? If Yes then the number is divisible by 3.

Number Sum of Digits Is divisible by 3? Result

78

7+8=15
Yes
As 15 is divisible by 3 So 78 is divisible
134
1+3+4=8
No
As 8 is not divisible by 3 So 134 is not divisible
261
2+6+1=9
Yes
9 is divisible by 3 So 261 is divisible
378
3+7+8=18
Yes
As 18 is divisible by 3 So 378 is divisible
6566
6+5+6+6=23
No
As 23 is not divisible by 3 SO 6566 is not divisible
46323
4+6+3+2+3= 18
Yes
As 18 is divisible by 3 So 46323 is divisible by 3

### Divisibility by 4

A number is divisible by 4 if its last two digits from your right hand side is divisible by 4.

Number Last Two Digit Is divisible by 4? Result

156

56
Yes
As 56 is divisible by 4 So 156 is divisible
242
42
No
As 42 is not divisible by 4 So 242 is not divisible
456368
68
Yes
As 68 is divisible by 4 So 456368 is divisible by 4

### Divisibility by 5

If the unit digit of any number is either 0 or 5 then we can say that the number is divisible by 5. Otherwise the number is not divisible by 5.

Number Last Digit Is divisible? Result

650

0
Yes
As the unit digit is 0 So 650 is divisible by 5.
56002
2
No
As Unit digit is not 0 or 5 so it is not divisible.
79856385
5
Yes
As the unit digit is 5 So 79856385 is divisible by 5

### Divisibility by 6

We can check that a number is divisible by 6 only if the number is divisible by both 2 and 3. That means we have to apply the divisibility rule of 2 and divisibility rule of 3 to and check whether the number is divisible by both or not and it must be an even number.If not the number is not divisible by 6.

A number is divisible by 6 if it is an even number and divisible by 3.

Number Is divisible by 2? Is divisible by 3? Result

96

Yes
Yes
As the number is divisible by 2 and 3 the number is divisible by 6.
256
Yes
No
The number is divisible by 2 but ot divisible by 3 So 256 is not divisible by 6
263
No
Yes
As it is an odd number it is not divisible by 6
1266
Yes
Yes
As 1266 is divisible by 2 and 3 so it is divisible by 6.

### Divisibility by 7

To check the divisibility rule of 7 we have to perform the following steps:

1. Multiply the unit digit by 2.
2. Now subtract the multiplication result from the remaining digits.
3. If the subtraction result is divisible by 7 then the number is divisible by 7.

Note: Use this rule till 5 digit Number.

Number Unit Digit Double of Unit Digit Subtraction Result

161

1

1 × 2= 2

(16)1= 16-2=14
As 14 is divisible by So 161 is divisible by 7
306

6

6 × 2= 12
(30)6= 30-6=24
As 24 is not divisible by 7 So 306 is not divisible by 7
693
3
3 × 2= 6
(69)3= 69-6= 63
As 63 is divisible by 7 So 693 is divisible by 7

If the number exceeds 5 digits, then do not use the above divisibility rule. Because it will be difficult for you to check the divisibility.

#### Divisibility Rule of 7 more than 5 Digit Number

1. First, make pairs of 3 digits from your right hand side.
3. Subtract the addition result pairs

Lets understand by taking Examples

Q. Check whether 67666662 is divisible by 7 or not?

Step-1: Make Pair of 3 digits from right side (67) -(666)-(662)

Step-2: Add Alternative Pairs 67+662= 729

Step-3: Substract from the remaining pairs. 729-666=63

Here 63 is divisible by 7 So 6766662 is divisible by 7

### Divisibility by 8

If the last 3 digits from your right hand side is divisible by 8 then then the number is divisible by 8.

One initial check is that the number must be an even number, otherwise the number is not divisible by 8.

Number Last Three Digit Is divisible by 8? Result

263152

152
Yes
As 152 is divisible by 8 So 263152 is divisible
566364
364
No
As 364 is not divisible by 4 So 566364 is not divisible
5663368
368
Yes
As 368 is divisible by 8 So 5663368 is divisible by 4

### Divisibility by 9

A number is divisible by 9 only if the sum total of all the digits is divisible by 9. We have to add all the digits like the divisibility rule of 3 and check whether it is divisible by 9 or not.

Number Sum of Digit Is divisible by 9? Result

6363

6+3+6+3=18
Yes
As Sum 18 is divisible by 9 SO the number is divisible by 9
96364
9+6+3+6+4=28
No
As sum is 28 which is not divisible by 9 So the number is not divisible by 9.

### Divisibility by 10

Any number which has the only unit digit 0 is divisible by 10.

Number Unit Digit Is divisible by 10? Result

62610

0
Yes
As its unit digit is 0 so it is divisible by 10
554545
5
No
As its unit digit is 5 so it is not divisible by 10

### Divisibility by 11

A number is diviosible by 11 if the difference between the sum of even place and sum of odd places are divisible 11 or 0.

Number Sum of Even Places Sum of Odd Places Odd-Even Result

244354

5+4+2=11
4+3+4=11
11-11=0
As the difference is 0 so the number is divisible by 11
206140
2+6+4=12
0+1+0=1
12-1=11
As the diference is 11 which is divisible by 11 SO the number is divisible by 11
636652
5+6+6=17
2+6+3=11
11-17=-6
As -6 is not divisible by 11 so the number is not divisible by 11

# HCF and LCM Concept Shortcut Trick

Hey Guys, Are you facing difficulty while solving HCF and LCM problems? Don’t worry, just follow the post and  your all difficulties and doubt will be cleared. Here we have mentioned the basic concept of HCF and LCM along with the shortcut tricks to solve all problems.

### Why you should learn HCF and LCM?

If you want to be strong in Mathematics and Aptitude, then you have to be strong in HCF and LCM. HCF and LCM are two backbones of Mathematics.  As like you need food to make your body function similarly you have to be strong in LCM and HCF to do solve further mathematics. Because you will use this concept in every step of mathematics like Addition, Substraction, Calculus and other solutions.

### What is the concept of Factor?

A bunch of numbers multiplied with each other to form a new number is called factor of that new number.

#### How to calculate the factor of a number?

Let us understand the concept of factor by taking an example:

Example: Find the factor of 24, 25, 28?

Solution: (i) 24= 2 × 2 × 2 × 3

(ii) 25= 5 × 5

(iii) 28 = 2 × 2 × 7

I think the concept of factor is now cleared.

### HCF (Highest Common Factor)

HCF of two or more number is a number which divides each and every number exactly. In genral, we can say that the highest common factors between two or more numbers is called HCF of that number.

#### Common Factor

Common factor is the factors which are  present in both the number is called common factor.

Example: Factor of 6= 2, 3

& 12= 2, 2, 3

So we can say that 2 and 3  are the common factor of both the number.

Example of HCF of two numbers

Q. Calculate the HCF of 6 and 12

Factor of 6= 2 , 3

Factor of 12= 2, 2, 3

Here Common factors are 2 and 3

So HCF = 2 × 3=6

Q. Find the HCF of 6 and 14?

### LCM (Least common multiple)

LCM is a least number which is divisible by all the given numbers. In other words, we can say that LCM is the product of highest factor of a given number.

Example: LCM of 6, 8

6= 2, 3

8= 2, 2, 2

So LCM= 2 × 2 × 2 × 3= 24

Example 2: LCM of 6 and 14?

### Important Formulas

• HCF × LCM = Product of numbers

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### HCF and LCM of Decimal Number

Lets take an example to understand How to find the HCF and LCM of Decimal number.

Q. Find the GCD and LCM of the number 0.6, 0.18 and 1.2?

Step-1: First of all check what is the number of digits after  ‘.’ (Dot) point.

Step-2: Convert all the digits after ‘.’ point into two digits. If one digit is present, put ‘0’ to make it two digits.

Step-3: Remove ‘.’ dot from all the number.

60       18     120

Step-4: Calculate HCF of 60, 18, 120

Step-5: Calculate LCM of 60, 18, 120

So LCM= 2 × 3 × 2 × 5 × 3 × 2=360

Step-6: Divide the calculated result by 100 to calculate HCF and LCM of the given numbers.

HCF= 6/100=0.06

LCM= 360/100= 3.60

### HCF and LCM of Fraction

$\fn_jvn&space;HCF=&space;\frac{HCF&space;of&space;(Numerator)}{LCM&space;of&space;(Denominator)}$

$\fn_jvn&space;LCM=&space;\frac{LCM&space;of&space;(Numerator)}{HCF&space;of&space;(Denominator)}$

Q. Find the LCM and GCD of  $\fn_jvn&space;\frac{6}{21},&space;\frac{8}{35}&space;and&space;\frac{12}{63}$ ?

Solution:

$\fn_jvn&space;LCM=&space;\frac{LCM(&space;6,&space;8,&space;12)}{HCF(&space;21,&space;35,&space;63)}=&space;\frac{24}{7}$

$\fn_jvn&space;HCF&space;=&space;\frac{HCF(6,&space;8,&space;12&space;)}{LCM(2,&space;35,&space;63&space;)}=&space;\frac{2}{315}$

### HCF and LCM of Power of a Number

Q. Find the GCM and LCM of 6², 6¹³, 6 ^18, 6^19? (^=Power)

Solution: HCF= 6²

LCM = 6^19

Q. Find the LCM and GCM of 3^-2 , 3^-12, 3^-23, 3 ^-32? (^=Power)

Solution: HCF= 6²

LCM = 6^19

### HCF and LCM of Power of Polynomial

Q. What is the LCM and HCF of 2ab, 6a²b, 8a²b² ?

Solution: Given Expression= 2ab, 6a²b, 8a²b²

We can write 2ab, 6a²b, 8a²b²= 2ab, 2 × 3 a²b, 2³a²b²

So our final HCF= 2ab

and LCM= 2³a²b²

Q. What is the LCM and HCF of x² – xy, x – y, 3x – 3y, x²- 2xy + y² ?

Solution: Given Expression= x² – xy, x – y, 3x – 3y, x²- 2xy + y

We can write  as x( x – y), x – y, ( x – y)²

So our final HCF= x – y

and LCM= x ( x-y)²

I think your concept regarding this topic is cleared. If something is missing in this post, then kindly let me know about that.

# Number system Questions with Answers

Number System Questions with Solution: Hey Guys, Do you know, the number system questions are one of the cool problems in the Aptitude section of Mathematics. You just have to add, substract, multiply or divide one number with another number. I think you are very much aquented with the number system problems. So lets discuss  different problems of number system with their solutions.

## Number System Questions with Solutions

Q.1 What will be the solution of 34598205 × 999=?

(A) 34563606747
(B) 34563606795
(C) 34563606022
(D) 34563444222

Solution:Here we have two  process to solve this question. First way we can multiply 34598205 with 999 which will be lengthy. So we have to follow the second method i.e:

34598205 × ( 1000 - 1 )

=34598205000 - 34598205

= 34563606795

Q.2 What is the value of 998 × 112 + 998 × 88= ?

(A) 199600
(B) 156666
(C) 456221
(D) 196632

Tips: To solve this question we have to use distributive law:

Solution:  998 × 112 + 998 × 88= 998 × ( 112 + 88) =998 × 200 = 199600

Related Topics:

Q.3 What is the value of

742 × 742 + 258 × 258 + 2 × 742 × 258= ?

(A) 10000000
(B) 45668222
(C) 1000000
(D) 4522588

Tips: Use ( a + b )²= a² + b² + 2ab

Solution: 742 × 742 + 258 × 258 + 2 × 742 × 258= (742)² + (258)² + 2 × 742 × 258= (742 + 258)²=1000²=1000000

Q.4 Find the value of  $for which 236$62 is divisible by 3 and must be an odd number?
(A) 3
(B) 5
(C) 7
(D) 9

Tips: Use the divisible formula for 3 i.e A number is divisible by 3 if the sum of all the digit of the number is divisible by 3.

Solution: Sum of the given number 2 + 3 + 6  + 6 + 2= 18

Q.5 Weather 876744 is divisible by 88?
(A) Yes
(B) No

Tips: Use divisible formula of 11 and 8.

Solution: 88= 11 × 8

• Sum of digits at odd place - Sum of digits at even place = ( 8 + 6 + 4) - ( 7 + 7 + 4)= 18 - 18= 0 which is divisible by 11 So 876744 is divisible by 11
• Last 3 digit of 876744 is 744 which is divisible by 8, so 876744 is divible by 8.

So, 876744 is divisible by 88 because 876744 is divisible by both 11 and 8.

Q.6 What will be the unit digit in the multiplication of (5227)³¹³ ×(91)³²¹?
(A) 1
(B) 3
(C) 7
(D) 9

Solution: We have to calculate the unit digit in (5227)³¹³ × (91)³²¹

As we know 74 having 1 as unit digit and (5227)³¹³ 's unit digit can be calculated as per the power of 7 and (91)³²¹ unit digit can be calculated by the help of 1.

So we can write (7)³¹³× (1)³²¹=(7)³¹² × 7 × (1)³²¹= ((7)4)78  × 7 × 1

Hence our required unit digit = ( 1 × 7 × 1) = 7

Q.7  What is the minimum number we have to add with 2019 to obtain a number which is completely divisible by 11?

(A) 12
(B) 5
(C) 6
(D) 4

Tips: Use formula  Required Number = Divisor - Remainder

Solution:

So our Required Number= 11 - 6 = 5

Q.8  What is the minimum number we have to subtract with 2019 to obtain a number that is completely divisible by 11?

(A) 12
(B) 5
(C) 6
(D) 4

Tips: Use formula  Required Number =Remainder

Solution:

So Our Required Number=6

Q.9 What will be the sum of 1² + 2 ² + 3² + 4² + 5²+ ……..+ 890² ?

(A) 396494
(B) 396495
(C) 396496
(D) 396497

Tips: Use the series formula  1² + 2 ² + 3² + 4² + 5²+ ........+ n²= ½n(n+1) (2n + 1)

Solution: In 1² + 2 ² + 3² + 4² + 5²+ ........+ 890² here n= 890

So as per the formula we have: ½ × 890(890+1) (2×890 + 1) = 445 × 891 × 1780= 705761100

Q.10 What will be the sum of 1³ + 2³ + 3³ + 4³ + 5³+ ……..+18³​ ?

(A) 705761101
(B) 128608721
(C) 128608722
(D) 12860855