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**Q.1. Test the divisibility of following numbers by 2 :****(i) 2650****(ii) 69435****(iii) 59628****(iv) 789403****(v) 357986****(vi) 367314****Ans.****(i)** The given number = 2650

Digit at unit’s place = 0

∴ It is divisible by 2.**(ii)** The given number = 69435

Digit at unit’s place = 5

∴ It is not divisible by 2.**(iii)** The given number = 59628

Digit at unit’s place = 8

∴ It is divisible by 2.**(iv)** The given number = 789403

Digit at unit’s place = 3

∴ It is not divisible by 2.**(v) **The given number = 357986

Digit at unit’s place = 6

∴ It is divisible by 2.**(vi)** The given number = 367314

Digit at unit’s place = 4

∴ It is divisible by 2.**Q.2. Test the divisibility of following numbers by 3 :****(i) 733****(ii) 10038****(iii) 20701****(iv) 524781****(v) 79124 ****(vi) 872645****Ans.****(i) **The given number = 733

Sum of its digits = 7 + 3 + 3 = 13, which is not divisible by 3.

∴ 733 is not divisible by 3.**(ii)** The given number = 10038

Sum of its digits = 1 + 0 + 0 + 3 + 8

= 12, which is divisible by 3

∴ 10038 is divisible by 3.**(iii) **The given number = 20701

Sum of its digits = 2 + 0 + 7 + 0 + 1

= 10, which is not divisible by 3

∴ 20701 is not divisible by 3.**(iv)** The given number = 524781 Sum of its digits = 5 + 2 + 4 + 7 + 8 + 1 = 27, which is divisible by 3

∴ 524781 is divisible by 3.**(v)** The given number = 79124 Sum of its digits = 7 + 9 + 1 + 2 + 4 = 23, which is not divisible by 3

∴ 79124 is not divisible by 3.**(vi)** The given number = 872645 Sum of its digits = 8 + 7 + 2 + 6 + 4 + 5 = 32, which is not divisible by 3

∴ 872645 is not divisible by 3.**Q.3. Test the divisibility of the follo wing numbers by 4 :****(i) 618****(ii) 2314****(iii) 63712****(iv) 35056****(v) 946126****(vi) 810524****Ans.** **(i) **The given number = 618

The number formed by ten’s and unit’s digits is 18, which is not divisible by 4.

∴ 618 is not divisible by 4.**(ii)** The given number = 2314

The number formed by ten’s and unit’s

digits is 14, which is not divisible by 4.

∴ 2314 is not divisible by 4.**(iii)** The given number = 63712

The number formed by ten’s and unit’s digits is 12, which is divisible by 4

∴ 63712 is divisible by 4.**(iv) **The given number = 35056

The number formed by ten’s and unit’s digits is 56, which is divisible by 4.

∴ 35056 is divisible by 4.**(v)** The given number = 946126

The number formed by ten’s and unit’s digits is 26, which is not divisible by 4.

∴ 946126 is not divisible by 4.**(vi)** The given number = 810524

The number formed by ten’s and unit’s digits is 24, which is divisible by 4.

∴ 810524 is divisible by 4.**Q.5. Test the divisibility of the follo wing numbers by 6 :****(i) 2070****(ii) 46523****(iii) 71232****(iv) 934706****(v) 251730****(vi) 9087248****Ans.****(i) **The given number = 2070

Its unit’s digit = 0

So, it is divisible by 2

Sum of its digits = 2 + 0 + 7 + 0 = 9,

which is divisible by 3

∴ The given number is divisible by 3.

So, 2070 is divisible by both 2 and 3.

Hence it is divisible by 6.**(ii) **The given number = 46523

Its unit’s digit = 3

So, it is not divisible by 2

Hence 46523 is not divisible by 6.**(iii)** The given number = 71232

Its unit’s digit = 2

So, it is divisible by 2

Sum of its digits = 7 + 1 + 2 + 3 + 2

= 15, which is divisible by 3

∴ 71232 is divisible by both 2 and 3

Hence it is divisible by 6.**(iv) **The given number = 934706

Its unit’s digit = 6 So,

it is divisible by 2

Sum of its digits = 9 + 3 + 4 + 7 + 0 + 6

= 29, which is not divisible by 3

Hence 934706 is not divisible by 6.**(v)** The given number = 251730

Its unit’s digit = 0

So, it is divisible by 2

Sum of its digits = 2 + 5 + 1 + 7 + 3 + 0

= 18, which is divisible by 3

∴ 251730 is divisible by both 2 and 3.

Hence it is divisible by 6.**(vi) **872536 is not divisible by 6 as sum of

its digits is 8 + 7 + 2 + 5 + 3 + 6 =31

which is not divisible by 3**Q.6 . Test the divisibility of the follo wing numbers by 7 :****(i) 826****(ii) 117****(iii) 2345****(iv) 6021****(v) 14126****(vi) 25368****Ans.** We know that a number is divisible by the difference between twice the ones digit and the number formed by the other digits is either 0 or a multiple of 7**(i)** 826, 6 × 2 = 12 and 82

Difference between 82 and 12 = 70

Which is divisible by 7

∴ 826 is divisible by 7**(ii)** In 117, 7 × 2 = 14, 11

Difference between 14 and 11 = 14 – 11

= 3

Which is not divisible by 7

∴ 117 is not divisible by 7**(iii)** In 2345, 5 × 2 = 10 and 234

Difference between 234 – 10 = 224

which is divisible by 7

∴ 2345 is divisible by 7**(iv) **In 6021, 1 × 2 = 2, and 602

Difference between 602 and 2 = 600

which is not divisible by 7

∴ 6021 is not divisible by 7**(v) **In 14126, 6 × 2 = 12 and 1412

Difference between 1412 – 12 = 1400

which is divisible by 7

∴ 14126 is divisible by 7**(vi)** In 25368, 8 × 2 = 16 and 2536

Difference between 2536 and 16 = 2520

which is divisible 7

∴ 25368 is divisible by 7**Q.7. Test the divisibility of the follo wing numbers by 8 :****(i) 9364****(ii) 2138****(iii) 36792****(iv) 901674****(v) 136976****(vi) 1790184****Ans.****(i) **The given number = 9364

The number formed by hundred’s, ten’s and unit’s digits is 364, which is not divisible by 8.

∴ 9364 is not divisible by 8.

**(ii)** The given number = 2138

The number formed by hundred’s, ten’s and unit’s digits is 138, which is not divisible by 8.

∴ 2138 is not divisible by 8.**(iii) **The given number = 36792

The number formed by hundred’s, ten’s and unit’s digits is 792, which is divisible by 8.

∴ 36792 is divisible by 8.**(iv)** The given number = 901674

The number formed by hundred’s, ten’s and unit’s digits is 674, which is not divisible by 8.

∴ 901674 is not divisible by 8.

( v ) The given number = 136976

The number formed by hundred’s, ten’s and unit’s digits is 976, which is divisible by 8.

∴ 136976 is divisible by 8.

(vi) The given number = 1790184

The number formed by hundred’s, ten’s and unit’s digits is 184, which is divisible by 8.

∴ 1790184 is divisible by 8.**Q.8. Test the divisibility of the following numbers by 9 :****(i) 2358****(ii) 3333****(iii) 98712****(iv) 257106****(v) 647514****(vi) 326999****Ans.**

We know that a number is divisible by 9, if the sum of its digits is divisible by 7**(i)** In 2358

Sum or digits : 2 + 3 + 5 + 8 = 18 which is divisible by 9

∴ 2358 is divisible by 9**(ii)** In 3333

Sum of digit 3 + 3 + 3 + 3 = 12 which is not divisible by 9

∴ 3333**(iii)** In 98712

Sum of digits = 9 + 8 + 7 + 1 + 2 = 27

Which is divisible by 9

∴ 98712 is divisible by 9**(iv) **In 257106

Sum of digits = 2 + 5 + 7 + 1 + 0 + 6 = 21 which is not divisible by 9

∴ 257106 is not divisible by 9**(v)** In 647514

Sum of digits = 6 + 4 + 7 + 5 + 1 + 4 = 27 which is divisible by 9

∴ 647514 is divisible by 9**(vi)** In 326999

Sum of digits = 3 + 2 + 6 + 9 + 9 + 9 = 38 which is not divisible by 9

∴ 326999 is divisible by 9**Q.9. Test the divisibility of the following number by 10 :****(i) 5790(ii) 63215**

∴ (i) 5790 is divisible by 10

**(v) 901351****(vi) 8790322****Ans.****(i)** The given number = 4334

Sum of its digits in odd places = 4 + 3 = 7

Sum of its digits in even places = 3 + 4 = 7

Difference of the two sums = 7 – 7 = 0

∴ 4334 is divisible by 11.**(ii)** The given number = 83721

Sum of its digits in odd places = 1 + 7 + 8 = 16

Sum of its digits in even places = 2 + 3 = 5

Difference of the two sums = 16 – 5 = 11, which is multiple of 11.

∴ 83721 is divisible by 11.**(iii)** The given number = 66311

Sum of its digits in odd places

= 1 + 3 + 6 = 10

Sum of its digits in even places

= 1 + 6 = 7

Difference of the two sums = 10 – 7 = 3, which is not a multiple of 11.

∴ 66311 is not divisible by 11.**(iv)** The given number = 137269

Sum of its digits in odd places

= 9 + 2 + 3 = 14

Sum of its digits in even places

= 6 + 7 + 1 = 14

Difference of the two sums

= 14 – 14 = 0

∴ 137269 is divisible by 11.**(v)** The given number = 901351

Sum of its digits in odd places

= 1 + 3 + 0 = 4

Sum of its digits in even places

= 5 + 1 + 9 = 15

Difference of the two sums = 15 – 4

= 11, which is a multiple of 11.

∴ 901351 is divisible by 11.

(vi) The given number = 8790322

Sum of its digits in odd places

= 2 + 3 + 9 + 8 = 22

Sum of its digits in even places

= 2 + 0 + 7 = 9

Difference of the two sums

= 22 – 9 = 13,

which is not a multiple of 11.

∴ 8790322 is not divisible by 11.**Q.11. In each of the following numbers , replace * by the smallest number to make it divisible by 3.****(i) 27*4****(ii) 53*46****(iii) 8*711****(iv) 62*35****(v) 234*17**

**(vi) 6*1054****Ans.**** (i) **The given number = 27*4

Sum of its digits = 2 + 7 + 4 = 13

The number next to 13 which is divisible by 3 is 15.

∴ Required smallest number = 15 – 13

= 2.**(ii)** The given number = 53*46

Sum of the given digits = 5 + 3 + 4 + 6

= 18, which is divisible by 3.

∴ Required smallest number = 0.**(iii)** The given number = 8*711

Sum of the given digits = 8 + 7 + 1 + 1 = 17

The number next to 17, which is divisible by 3 is 18.

∴ Required smallest number = 18 – 17

= 1.**(iv) **The given number = 62*35

Sum of the given digits = 6 + 2 + 3 + 5

= 16

The number next to 16, which is divisible by 3 is 18.

∴ Required smallest number = 18 – 16

= 2

**(v)** The given number = 234*17

Sum of the given digits

= 2 + 3 + 4 + 1 + 7 = 17

The number next to 17, which is divisible by 3 is 18.

∴ Required smallest number

= 18 – 17 = 1.**(vi) **The given number = 6* 1054

Sum of the given digits = 6 + 1 + 0 + 5 + 4 = 16

The number next to 16, which is divisible by 3 is 18.

∴ Required smallest number

= 18 – 16 = 2.**Q.12. In each of the following numbers, replace * by the smallest number to make it divisible by 9.****(i) 65*5****(ii) 2*135****(iii) 6702*****(iv) 91*67****(v) 6678*1****(vi) 835*86****Ans.****(i)** The given number = 65*5

Sum of its given digits = 6 + 5 + 5 = 16

The number next to 16, which is divisible by 9 is 18.

∴ Required smallest number = 18 – 16

= 2.**(ii) **The given number = 2*135

Sum of its given digits = 2 + 1 + 3 + 5

= 11

The number next to 11, which is divisible by 9 is 18.

∴ Required smallest number

= 18 – 11 = 7.**(iii)** The given number = 6702*

Sum of its given digits

= 6 + 7 + 0 + 2 = 15

The number next to 15, which is divisible by 9 is 18.

∴ Required smallest number = 18 – 15 = 3**(iv) **The given number = 91*67

Sum of its given digits = 9 + 1 + 6 + 7 = 23

The number next to 23, which is divisible by 9 is 27.

∴ Required smallest number = 27 – 23 = 4.**(v)** The given number = 6678*1

Sum of its given digits

= 6 + 6 + 7 + 8 + 1 = 28

The number next to 28, which is divisible by 9 is 36.

∴ Required smallest number

= 36 – 28 = 8.**(vi)** The given number = 835*86

Sum of its given digits

= 8 + 3 + 5 + 8 + 6

= 30

The number next to 30, which is divisible

by 9 is 36.

∴ Required smallest number

= 36 – 30 = 6.**Q.13. In each of the following numbers, replace * by the smallest number to make it divisible by 11.****(i) 26*5****(ii) 39*43****(iii) 86*72****(iv) 467*91****(v) 1723*4**

**(vi) 9*8071**

Ans. **(i)** The given number = 26*5

Sum of its digits is odd places

= 5 + 6 = 11

Sum of its digits in even places = * + 2

Difference of the two sums

= 11 – (* + 2)

The given number will be divisible by 11 if the difference of the two sums = 0.

∴ 11 – (* + 2) = 0

11 = * + 2

11 – 2 = *

9 = *

∴ Required smallest number = 9.**(ii)** The given number = 39*43

Sum of its digits in odd places

= 3 + * + 3 = * + 6

Sum of its digits in even places

= 4 + 9 = 13

Difference of the two sums

= * + 6 – 13 = * – 7

The given number will be divisible by 11,

if the difference of the two sums = 0.

∴ * – 7 = 0

* = 7

∴ Required smallest number = 7.**(iii)** The given number = 86*72

Sum of its digits in odd places

= 2 + * + 8 = * + 10

Sum of its digits in even places

= 7 + 6 = 13

Difference of the two sums

= * + 10 – 13 = * – 3

The given number will be divisible by

11, if the difference of the two sums = 0.

∴ * – 3 = 0

* = 3

∴ Required smallest number = 3.**(iv)** The given number = 467*91

Sum of its digits in odd places

= 1 + * + 6 = * + 7

Sum of its digits in even places

= 9 + 7 + 4 = 20

Difference of the two sums

= 20 – (* + 7)

= 20 – * – 7 = 13 – *

Clearly the difference of the two sums

will be multiple of 11 if 13 – * = 11

∴ 13 – 11 = *

2 = *

* = 2

∴ Required smallest number = 2.**(v)** The given number = 1723*4

Sum of its digits in odd places

= 4 + 3 + 7 = 14

Sum of its digits in even places

= * + 2 + 1 = * + 3

Difference of the two sums

= * + 3 – 14 = * – 11

∴ The given number will be divisible by

11, if * – 11 is a multiple of 11, which is

possible if * = 0.

∴ Required smallest number = 0.**(vi)** The given number = 9*8071

Sum of its digits in odd places

= 1 + 0 + * = 1 + *

Sum of its digits in even places

= 7 + 8 + 9 = 24

Difference of the two sums

= 24 – 1 – * = 23 – *

∴ The given number will be divisible by

11, if 23 – * is a multiple of 11, which is possible if * = 1.

∴ Required smallest number = 1.**Q.14. Test the divisibility of**

**(i) 10000001 by 11****(ii) 19083625 by 11****(iii) 2134563 by 9****(iv) 10001001 by 3****(v) 10203574 by 4**

**(vi) 12030624 by 8**

**Ans.****(i)** The given number = 10000001

Sum of its digits in odd places

= 1 + 0 + 0 + 0 = 1

Sum of its digits in even places

= 0 + 0 + 0 + 1 = 1

Difference of the two sums = 1 – 1 = 0

∴ The number 10000001 is divisible by 11.**(ii)** The given number = 19083625

Sum of its digits in odd places

= 5 + 6 + 8 + 9 = 28

Sum of its digits in even places

= 2 + 3 + 0 + 1 = 6

Difference of the two sums = 28 – 6

= 22, which is divisible by 11.

∴ The number 19083625 is divisible by 11.**(iii) **The given number = 2134563

Sum of its digits = 2 + 1 + 3 + 4 + 5 + 6 + 3

= 24, which is not divisible by 9.

∴ The number 2134563 is not divisible by 9.**(iv)** The given number = 10001001

Sum of its digits = 1 + 0 + 0 + 0 + 1 + 0 + 0 + 1 = 3, which is divisible by 3.

∴ The number 10001001 is divisible by 3.**(v)** The given number = 10203574

The number formed by its ten’s and unit’s digits is 74, which is not divisible by 4.

∴ The number 10203574 is not divisible by 4.

**(vi)** The given number = 12030624

The number formed by its hundred’s, ten’s and unit’s digits = 624, which is divisible by 8.

∴ The number 12030624 is divisible by 8.**Q.15. Which of the following are prime numbers ?****(i) 103****(ii)137****(iii) 161****(iv) 179****(v) 217****(vi) 277****(vii) 331****(viii) 397****Ans.**

103, 137, 179, 277, 331, 397 are prime numbers.**Q.16. Give an example of a number****(i) Which is divisible by 2 but not by 4.****(ii) Which is divisible by 4 but not by 8.****(iii) Which is divisible by both 2 and 8 but not divisible by 16.****(v) Which is divisible by both 3 and 6 but not by 18.****Ans.****(i) **154**(ii)** 612**(iii)** 5112, 3816**(iv)** 3426, 5142 etc.**Q.17. Write (T) for true and (F) for false against each of the followings statements :****(i) If a number is divisible by 4, it must be divisible by 8.**

**(ii) If a number is divisible by 8, it must be divisible by 4.****(iii) If a number divides the sum of two numbers exactly, it must exactly divide**

**the numbers separately.****(iv) If a number is divisible by both 9 and 10, it must be divisible by 90.****(v) A number is divisible by 18, if it is divisible by both 3 and 6.****(vi) If a number is divisible by 3 and 7, it must be divisible by 21.**

**(vii) The sum of two consecutive odd numbers is always divisible by 4.**

**(viii) If a number divides three numbers exactly, it must divide their sum exactly.****Ans.**

**(i) **False**(ii) **True**(iii)** False**(iv)** True**(v)** False

**(vi)** True

**(vii)** True**(viii)** True.**PRIME FACTORIZATION****Prime Factor.** A factor of a given number is called a prime factor if this factor is a prime number.**Prime Factorization.** To express a given number as a product of prime factors is called a prime factorization of the given number.

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