Playing With Numbers — Exercise 6(B)

Show that :

15 is a factor of 1065.

Answer

On dividing 1065 by 15, we get:

$\begin{array}{l} \phantom{22 -)}{71} \\ 15\overline{\smash{\big)}1065} \\ \phantom{22}\phantom{)}\mathllap{-,}\underline{105\phantom{+1}} \\ \phantom{{22 - 1}}15 \\ \phantom{22 - 1}\mathllap{-,}\underline{15} \\ \phantom{22 - 1)}\underline{0} \\ \end{array}$

On dividing 1065 by 15, the remainder is 0.

∴ 1065 is exactly divisible by 15.

Hence, 15 is a factor of 1065.

Show that :

17 is a factor of 1241.

Answer

On dividing 1241 by 17, we get:

$\begin{array}{l} \phantom{22 -)}{73} \\ 17\overline{\smash{\big)}1241} \\ \phantom{22}\phantom{)}\mathllap{-,}\underline{119\phantom{+1}} \\ \phantom{{22 - 1}}51 \\ \phantom{22 - 1}\mathllap{-,}\underline{51} \\ \phantom{22 - 1)}\underline{0} \\ \end{array}$

On dividing 1241 by 17, the remainder is 0.

∴ 1241 is exactly divisible by 17.

Hence, 17 is a factor of 1241.

Show that :

14 is not a factor of 2186.

Answer

On dividing 2186 by 14, we get:

$\begin{array}{l} \phantom{--}{156} \\ 14\overline{\smash{\big)}2186} \\ \phantom{22}\phantom{)}\mathllap{-,}\underline{14\phantom{+1}} \\ \phantom{1234.}78\ \phantom{22 - }\mathllap{-,}\underline{70\phantom{1}} \\ \phantom{12356.}86\ \phantom{22 - 1}\mathllap{-,}\underline{84\phantom{1}} \\ \phantom{22 -12 }\underline{2\phantom{1}} \\ \end{array}$

On dividing 2186 by 14, the remainder is 2.

∴ 2186 is not exactly divisible by 14.

Hence, 14 is not a factor of 2186.

Show that :

23 is not a factor of 2789.

Answer

On dividing 2789 by 23, we get:

$\begin{array}{l} \phantom{--}{121} \\ 23\overline{\smash{\big)}2789} \\ \phantom{22}\phantom{)}\mathllap{-,}\underline{23\phantom{+1}} \\ \phantom{1234.}48\ \phantom{22 - }\mathllap{-,}\underline{46\phantom{1}} \\ \phantom{12356.}29\ \phantom{22 - 1}\mathllap{-,}\underline{23\phantom{1}} \\ \phantom{22 -12 }\underline{6\phantom{1}} \\ \end{array}$

On dividing 2789 by 23, the remainder is 6.

∴ 2789 is not exactly divisible by 23.

Hence, 23 is not a factor of 2789.

Write down all the factors of :

(i) 40

(ii) 56

(iii) 63

(iv) 84

Answer

(i) We have,

40 = 1 × 40

40 = 2 × 20

40 = 4 × 10

40 = 5 × 8

Hence, possible factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40.

(ii) We have,

56 = 1 × 56

56 = 2 × 28

56 = 4 × 14

56 = 7 × 8

Hence, possible factors of 56 are 1, 2, 4, 7, 8, 14, 28, 56.

(iii) We have,

63 = 1 × 63

63 = 3 × 21

63 = 7 × 9

Hence, possible factors of 63 are 1, 3, 7, 9, 21, 63.

(iv) We have,

84 = 1 × 84

84 = 2 × 42

84 = 3 × 28

84 = 4 × 21

84 = 6 × 14

84 = 7 × 12

Hence, possible factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.

Write down :

(i) First six multiples of 7.

(ii) First seven multiples of 13.

(iii) First five multiples of 19.

Answer

(i) Multiples of 7 are obtained by multiplying 7 with natural numbers.

7 × 1 = 7

7 × 2 = 14

7 × 3 = 21

7 × 4 = 28

7 × 5 = 35

7 × 6 = 42

The first six multiples of 7 are 7, 14, 21, 28, 35, 42.

(ii) Multiples of 13 are obtained by multiplying 13 with natural numbers.

13 × 1 = 13

13 × 2 = 26

13 × 3 = 39

13 × 4 = 52

13 × 5 = 65

13 × 6 = 78

13 × 7 = 91

The first seven multiples of 13 are 13, 26, 39, 52, 65, 78, 91.

(iii) Multiples of 19 are obtained by multiplying 19 with natural numbers.

19 × 1 = 19

19 × 2 = 38

19 × 3 = 57

19 × 4 = 76

19 × 5 = 95

The first five multiples of 19 are 19, 38, 57, 76, 95.

Which of the following are prime numbers?

15, 19, 29, 37, 63, 91, 47, 51, 73, 87, 97

Answer

A prime number has exactly 2 factors, 1 and itself.

Hence, the prime numbers are 19, 29, 37, 47, 73, 97.

Write all prime numbers between 30 and 60.

Answer

The prime numbers between 30 and 60 are 31, 37, 41, 43, 47, 53, 59.

Write all prime numbers between 70 and 100.

Answer

The prime numbers between 70 and 100 are 71, 73, 79, 83, 89, 97.

Express each of the following as a product of prime factors :

(i) 105

(ii) 180

(iii) 420

(iv) 462

(v) 1035

(vi) 1197

(vii) 2535

(viii) 4641

Answer

(i) By successive division of 105 by prime factors, we get:

$\begin{array}{r|l} 3 & 105 \\ \hline 5 & 35 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array}$

Hence, 105 = 3 × 5 × 7.

(ii) By successive division of 180 by prime factors, we get:

$\begin{array}{r|l} 2 & 180 \\ \hline 2 & \phantom{1}90 \\ \hline 3 & \phantom{1}45 \\ \hline 3 & \phantom{1}15 \\ \hline 5 & \phantom{11}5 \\ \hline & \phantom{11} 1 \end{array}$

Hence, 180 = 2 × 2 × 3 × 3 × 5.

(iii) By successive division of 420 by prime factors, we get:

$\begin{array}{r|l} 2 & 420 \\ \hline 2 & 210 \\ \hline 3 & 105 \\ \hline 5 & \phantom{1}35 \\ \hline 7 & \phantom{11}7 \\ \hline & \phantom{11} 1 \end{array}$

Hence, 420 = 2 × 2 × 3 × 5 × 7.

(iv) By successive division of 462 by prime factors, we get:

$\begin{array}{r|l} 2 & 462 \\ \hline 3 & 231 \\ \hline 7 & \phantom{1}77 \\ \hline 11 & \phantom{1} 11 \\ \hline & \phantom{11} 1 \end{array}$

Hence, 462 = 2 × 3 × 7 × 11.

(v) By successive division of 1035 by prime factors, we get:

$\begin{array}{r|l} 3 & 1035 \\ \hline 3 & \phantom{1}345 \\ \hline 5 & \phantom{1}115 \\ \hline 23 & \phantom{11} 23 \\ \hline & \phantom{111} 1 \end{array}$

Hence, 1035 = 3 × 3 × 5 × 23.

(vi) By successive division of 1197 by prime factors, we get:

$\begin{array}{r|l} 3 & 1197 \\ \hline 3 & \phantom{1}399 \\ \hline 7 & \phantom{1}133 \\ \hline 19 & \phantom{11} 19 \\ \hline & \phantom{111} 1 \end{array}$

Hence, 1197 = 3 × 3 × 7 × 19.

(vii) By successive division of 2535 by prime factors, we get:

$\begin{array}{r|l} 3 & 2535 \\ \hline 5 & \phantom{1}845 \\ \hline 13 & \phantom{1}169 \\ \hline 13 & \phantom{11} 13 \\ \hline & \phantom{111} 1 \end{array}$

Hence, 2535 = 3 × 5 × 13 × 13.

(viii) By successive division of 4641 by prime factors, we get:

$\begin{array}{r|l} 3 & 4641 \\ \hline 7 & 1547 \\ \hline 13 & \phantom{1}221 \\ \hline 17 & \phantom{11} 17 \\ \hline & \phantom{111} 1 \end{array}$

Hence, 4641 = 3 × 7 × 13 × 17.

Fill in the blanks :

(i) ............... is a factor of every number.

(ii) ............... is a multiple of every number.

(iii) ............... is the only even prime number.

(iv) ............... is neither prime nor composite.

(v) The largest 2-digit prime number is ............... .

(vi) There are ............... prime numbers between 1 and 100.

Answer

(i) 1 is a factor of every number.

Reason: Every number is divisible by 1 without any remainder.

(ii) 0 is a multiple of every number.

Reason: When any number is multiplied by 0, the product is always 0.

(iii) 2 is the only even prime number.

Reason: 2 is an even number and is divisible only by 1 and 2. So, it is also a prime number.

(iv) 1 is neither prime nor composite.

Reason: A prime number has exactly two different factors, 1 and itself. A composite number has more than two factors. Since 1 has only one factor i.e. 1, it does not fit either definition.

(v) The largest 2-digit prime number is 97.

Reason: 97 has only two factors, 1 and 97, and there is no larger 2-digit prime number.

(vi) There are 25 prime numbers between 1 and 100.

Reason: By using the Sieve of Eratosthenes method to cross out multiples of 2, 3, 5, and 7, we are left with exactly 25 numbers that have no other factors besides 1 and themselves.

Which of the following numbers are divisible by 3 ?

(i) 57432

(ii) 693503

(iii) 3002002

(iv) 777771

Answer

Rule: A number is divisible by 3, only if the sum of its digits is divisible by 3.

(i) The sum of the digits of the number 57432 is,

5 + 7 + 4 + 3 + 2 = 21

Since the sum of the digits, 21, is divisible by 3, the number 57432 is also divisible by 3.

(ii) The sum of the digits of the number 693503 is,

6 + 9 + 3 + 5 + 0 + 3 = 26

Since the sum of the digits, 26, is not divisible by 3, the number 693503 is not divisible by 3.

(iii) The sum of the digits of the number 3002002 is,

3 + 0 + 0 + 2 + 0 + 0 + 2 = 7

Since the sum of the digits, 7, is not divisible by 3, the number 3002002 is not divisible by 3.

(iv) The sum of the digits of the number 777771 is,

7 + 7 + 7 + 7 + 7 + 1 = 36

Since the sum of the digits, 36, is divisible by 3, the number 777771 is also divisible by 3.

Which of the following numbers are divisible by 9?

(i) 666657

(ii) 999083

(iii) 756432

(iv) 876869

Answer

Rule: A number is divisible by 9, only if the sum of its digits is divisible by 9.

(i) The sum of the digits of the number 666657 is,

6 + 6 + 6 + 6 + 5 + 7 = 36

Since the sum of the digits, 36, is divisible by 9, the number 666657 is also divisible by 9.

(ii) The sum of the digits of the number 999083 is,

9 + 9 + 9 + 0 + 8 + 3 = 38

Since the sum of the digits, 38, is not divisible by 9, the number 999083 is not divisible by 9.

(iii) The sum of the digits of the number 756432 is,

7 + 5 + 6 + 4 + 3 + 2 = 27

Since the sum of the digits, 27, is divisible by 9, the number 756432 is also divisible by 9.

(iv) The sum of the digits of the number 876869 is,

8 + 7 + 6 + 8 + 6 + 9 = 44

Since the sum of the digits, 44, is not divisible by 9, the number 876869 is not divisible by 9.

Which of the following numbers are divisible by 11 ?

(i) 22222

(ii) 3303033

(iii) 1589148

(iv) 1302444

Answer

Rule: A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or a multiple of 11.

(i) Consider the number 22222.

Sum of its digits at odd places = 2 + 2 + 2 = 6

Sum of its digits at even places = 2 + 2 = 4

Their difference = 6 - 4 = 2

Since 2 is not 0 nor a multiple of 11, the number 22222 is not divisible by 11.

(ii) Consider the number 3303033.

Sum of its digits at odd places = 3 + 0 + 0 + 3 = 6

Sum of its digits at even places = 3 + 3 + 3 = 9

Their difference = 9 - 6 = 3

Since 3 is not 0 nor a multiple of 11, the number 3303033 is not divisible by 11.

(iii) Consider the number 1589148.

Sum of its digits at odd places = 8 + 1 + 8 + 1 = 18

Sum of its digits at even places = 4 + 9 + 5 = 18

Their difference = 18 - 18 = 0

Since the difference is 0, the number 1589148 is divisible by 11.

(iv) Consider the number 1302444.

Sum of its digits at odd places = 4 + 4 + 0 + 1 = 9

Sum of its digits at even places = 4 + 2 + 3 = 9

Their difference = 9 - 9 = 0

Since the difference is 0, the number 1302444 is divisible by 11.

Which of the following numbers are divisible by 4 ?

(i) 36518

(ii) 579012

(iii) 236754

(iv) 907246

Answer

Rule: A number is divisible by 4 if the number formed by its last two digits (tens and ones digits) is divisible by 4.

(i) Consider the number 36518. The number formed by its last two digits is 18 which is not divisible by 4.

Hence, the number 36518 is not divisible by 4.

(ii) Consider the number 579012. The number formed by its last two digits is 12 which is clearly divisible by 4.

Hence, the number 579012 is divisible by 4.

(iii) Consider the number 236754. The number formed by its last two digits is 54 which is not divisible by 4.

Hence, the number 236754 is not divisible by 4.

(iv) Consider the number 907246. The number formed by its last two digits is 46 which is not divisible by 4.

Hence, the number 907246 is not divisible by 4.

Which of the following numbers are divisible by 8 ?

(i) 13756

(ii) 467132

(iii) 6572904

(iv) 972972

Answer

Rule: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

(i) Consider the number 13756. The number formed by its last three digits is 756 which is not divisible by 8.

So the number 13756 is not divisible by 8.

(ii) Consider the number 467132. The number formed by its last three digits is 132 which is not divisible by 8.

So the number 467132 is not divisible by 8.

(iii) Consider the number 6572904. The number formed by its last three digits is 904 which is divisible by 8.

So the number 6572904 is divisible by 8.

(iv) Consider the number 972972. The number formed by its last three digits is 972 which is not divisible by 8.

So the number 972972 is not divisible by 8.

Which of the following numbers are divisible by 6 ?

(i) 835704

(ii) 367038

(iii) 630663

(iv) 626266

Answer

Rule: A number is divisible by 6 if it is divisible by both 2 and 3.

(i) Consider the number 835704.

Its unit digit is 4. So, it is divisible by 2.

The sum of its digits, 8 + 3 + 5 + 7 + 0 + 4 = 27, which is divisible by 3.

As the number 835704 is divisible by both 2 and 3, it is divisible by 6.

Hence, 835704 is divisible by 6.

(ii) Consider the number 367038.

Its unit digit is 8. So, it is divisible by 2.

The sum of its digits, 3 + 6 + 7 + 0 + 3 + 8 = 27, which is divisible by 3.

As the number 367038 is divisible by both 2 and 3, it is divisible by 6.

Hence, 367038 is divisible by 6.

(iii) Consider the number 630663.

Its unit digit is 3. So, it is not divisible by 2.

As the number 630663 is not divisible by 2, it is not divisible by 6.

Hence, 630663 is not divisible by 6.

(iv) Consider the number 626266.

Its unit digit is 6. So, it is divisible by 2.

The sum of its digits, 6 + 2 + 6 + 2 + 6 + 6 = 28, which is not divisible by 3.

As the number 626266 is not divisible by 3, it is not divisible by 6.

Hence, 626266 is not divisible by 6.

Which of the following numbers are divisible by 7 ?

(i) 67102

(ii) 73056

(iii) 102543

(iv) 203647

Answer

Rule: A number is divisible by 7 if the difference between double the digit at ones place and the number formed by the rest of its digits is divisible by 7.

(i) Consider the number 67102.

6710 - (2 × 2) = 6710 - 4 = 6706

On dividing 6706 by 7, we get 958 with remainder 0.

Thus, 6706 is divisible by 7.

∴ 67102 is also divisible by 7.

(ii) Consider the number 73056.

7305 - (2 × 6) = 7305 - 12 = 7293

On dividing 7293 by 7, we get 1041 with remainder 6.

Thus, 7293 is not divisible by 7.

∴ 73056 is also not divisible by 7.

(iii) Consider the number 102543.

10254 - (2 × 3) = 10254 - 6 = 10248

On dividing 10248 by 7, we get 1464 with remainder 0.

Thus, 10248 is divisible by 7.

∴ 102543 is also divisible by 7.

(iv) Consider the number 203647.

20364 - (2 × 7) = 20364 - 14 = 20350

On dividing 20350 by 7, we get 2907 with remainder 1.

Thus, 20350 is not divisible by 7.

∴ 203647 is also not divisible by 7.

Which of the following numbers are divisible by 5 ?

(i) 55513

(ii) 689015

(iii) 37490

(iv) 1010106

Answer

Rule: A number is divisible by 5 if its last digit is 0 or 5.

(i) The last digit of the number 55513 is 3.

Hence, 55513 is not divisible by 5.

(ii) The last digit of the number 689015 is 5.

Hence, 689015 is divisible by 5.

(iii) The last digit of the number 37490 is 0.

Hence, 37490 is divisible by 5.

(iv) The last digit of the number 1010106 is 6.

Hence, 1010106 is not divisible by 5.

Separate even and odd numbers out of the following :

(i) 267890

(ii) 357941

(iii) 668427

(iv) 200593

(v) 246825

(vi) 489562

Answer

Rule: Even numbers end in 0, 2, 4, 6, 8 whereas odd numbers end in 1, 3, 5, 7, 9.