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Divisibility Rules For 8 9 10 And 11

Divisibility Rule by 8 

A number is divisible by 8 if the last three digits of that number are divisible by 8 or a multiple of 8 like 008, 016, 024, 808, 816, 1168 .....

For example

  • Consider a number, 20240.

The last three digits, i.e. 240 are divisible by 8

240 \(\div\) 8

= 30

So, the number 20240 is divisible by 8.

  • Now consider another number, 80324.

The last three digits, i.e. 324 are not divisible by 8.

So, the number 80324 is not divisible by 8.

Illustration Questions

Which one of the following numbers is divisible by 8?

A 7032

B 3270

C 6025

D 8891

×

In the number 7032, the last three digits, i.e. 032 are divisible by 8.

032 \(\div\) 8 = 4

So, the number 7032 is divisible by 8.

In the number 3270, the last three digits, i.e. 270 are not divisible by 8.

So, the number 3270 is not divisible by 8.

In the number 6025, the last three digits, i.e. 025 are not divisible by 8.

So, the number 6025 is not divisible by 8.

In the number 8891, the last three digits, i.e. 891 are not divisible by 8.

So, the number 8891 is not divisible by 8.

Hence, option (A) is correct.

Which one of the following numbers is divisible by 8?

A

7032

.

B

3270

C

6025

D

8891

Option A is Correct

Divisibility Rule of 9

A number is divisible by 9 when the sum of all the digits of the number is a multiple of 9 like 9, 18, 27, 36, .....

For example:

  • Consider the number, 43506

The sum of all the digits = 4 + 3 + 5 + 0 + 6 = 18

18 is a multiple of 9, so the whole number 43506 is divisible by 9.

  • Now consider another number, 32505

The sum of all the digits = 3 + 2 + 5 + 0 + 5

= 15

15 is not a multiple of 9, so the whole number 32505 is not divisible by 9.

Illustration Questions

Which one of the following numbers is divisible by 9?

A 3012

B 3636

C 4531

D 4435

×

For the number 3012:

The sum of all the digits = 3 + 0 + 1 + 2 = 6

6 is not a multiple of 9, so the number 3012 is not divisible by 9.

For the number 3636:

The sum of all the digits = 3 + 6 + 3 + 6 = 18

18 is a multiple of 9, so the number 3636 is divisible by 9.

For the number 4531:

The sum of all the digits = 4 + 5 + 3 + 1 = 13

13 is not a multiple of 9, so the number 4531 is not divisible by 9.

For the number 4435:

The sum of all the digits = 4 + 4 + 3 + 5 = 16

16 is not a multiple of 9, so the number 4435 is not divisible by 9.

Hence, option (B) is correct.

Which one of the following numbers is divisible by 9?

A

3012

.

B

3636

C

4531

D

4435

Option B is Correct

Divisibility Rule of 10

A number is divisible by 10 if the last digit of the number is zero.

For example:

  • Consider the number, 2920

The last digit of this number is zero, so 2920 is divisible by 10.

  • Now consider another number, 1525

The last digit of this number is 5, so the number 1525 is not divisible by 10.

Illustration Questions

Which one of the following numbers is divisible by 10?

A 1998

B 1990

C 1997

D 1999

×

In the number 1998, the last digit is not zero, so 1998 is not divisible by 10.

In the number 1990, the last digit is zero, so 1990 is divisible by 10.

In the number 1997, the last digit is not zero, so 1997 is not divisible by 10.

In the number 1999, the last digit is not zero, so 1999 is not divisible by 10.

Hence, option (B) is correct.

Which one of the following numbers is divisible by 10?

A

1998

.

B

1990

C

1997

D

1999

Option B is Correct

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of the digits at odd positions and the sum of the digits at even positions is zero or a multiple of 11, like 11, 22, 33 .....

For example:

  1. Consider the number, 29271.

The sum of the digits at odd positions:

2 + 2 + 1 = 5

The sum of the digits at even positions:

9 + 7 = 16

The difference of the sum = 16 – 5 = 11

11 is a multiple of 11, so the number 29271 is divisible by 11.

      2. Now consider another number, 32312.

The sum of the digits at odd positions:

3 + 3 + 2 = 8

The sum of the digits at even positions:

2 + 1 = 3

The difference of the sum = 8 – 3  = 5

5 is not a multiple of 11, so the number 32312 is not divisible by 11.

      3. Consider another number, 4664.

The sum of the digits at odd positions:

4 + 6 = 10

The sum of the digits at even positions:

6 + 4= 10

Difference of the sum  = 10 – 10 = 0 

So, the number 4664 is divisible by 11.

Illustration Questions

Which one of the following numbers is divisible by 11?

A 511

B 671

C 777

D 111

×

For the  number 511:

The sum of the digits at odd positions = 5 + 1 = 6

The sum of the digits at even positions  = 1

The difference of the sum = 6 – 1 = 5

5 is not a multiple of 11, so 511 is not divisible by 11.

For the number 671:

The sum of the digits at odd positions = 6 + 1 = 7

The sum of the digits at even positions = 7

The difference of the sum = 7 – 7 = 0

So, it is divisible by 11.

For the number 777:

The sum of the digits at odd positions = 7 + 7 = 14

The sum of the digits at even positions = 7

The difference of the sum = 14 – 7 = 7

7 is not a multiple of 11, so 777 is not divisible by 11.

For the number 111:

The sum of the digits at odd positions = 1 + 1 = 2

The sum of the digits at even positions = 1

The difference of the sum = 2 – 1 = 1

1 is not a multiple of 11, so 111 is not divisible by 11.

Hence, option (B) is correct.

Which one of the following numbers is divisible by 11?

A

511

.

B

671

C

777

D

111

Option B is Correct

Relation among Divisor, Quotient, Remainder and Dividend

\(\text{Quotient × Divisor + Remainder = Dividend}\)

This relation is used to find the unknown term if the other three terms are given.

For example:

When \(28\) is divided by a number gives the quotient \(9\) and the remainder \(1.\) What is the number?

\(9×\) Divisor \(+1=28\)

Divisor \(=\dfrac{28-1}{9}\)

\(=\dfrac{27}{9}\)

\(=3\)

Illustration Questions

Alex divided a number by \(20\) and got the quotient of \(28\) with remainder as \(14\). What is the number?

A \(560\)

B \(570\)

C \(574\)

D \(564\)

×

Given:

Quotient \(=28\)

Remainder \(=14\)

Divisor \(=20\)

We have a relation, i.e.

\(\text{Quotient × Divisor + Remainder = Dividend}\)

On putting the given values, we get:

\(28×20+14=\) Dividend

Dividend \(=560+14\)

\(=574\)

Hence, option (C) is correct.

Alex divided a number by \(20\) and got the quotient of \(28\) with remainder as \(14\). What is the number?

A

\(560\)

.

B

\(570\)

C

\(574\)

D

\(564\)

Option C is Correct

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