3

3. Playing With Numbers

dA factor of a number is an exact divisor of that number.
dA number is said to be a multiple of any of its factors.

e.g., We know that 21 = 1 × 21 and 21 = 3 × 7.

This shows that each of the numbers 1, 3, 7 and 21 divides 21 exactly.

Therefore 1, 3, 7 and 21 are all factors of 21 and 21 is a multiple of each one of the
numbers 1, 3, 7 and 21.

dAll multiples of 2 are called even numbers.

e.g., 2, 4, 6, 8, 10, etc.,

dNumbers which are not multiples of 2 are called odd numbers.

e.g., 1, 3, 5, 7, 9, 11, etc., l1 is neither prime nor composite. Q
(Since 1 = 1, the two factors are U
dEach of the numbers which has exactly not distinct.) I
two distinct factors, namely 1 and itself C
is called a prime number. K

e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 etc., l2 is the lowest prime number.

dNumber having more than two factors is l2 is the only even prime number. TIP
known as composite number.
(All other even numbers are
e.g., 4, 6, 8, 9, 10 etc.
composite numbers.)

dTwo consecutive prime numbers differing by 2 are known as twin-primes.

e.g., (i) 3, 5 (ii) 5, 7 (iii) 11, 13 etc.,

A set of three consecutive prime numbers, differing by 2, is called a prime triplet.

An example of prime triplet is (3, 5, 7).

dIf the sum of all the factors of a number is twice the number then the number is
called a perfect number.

e.g., 6 is a perfect number, since the factors of 6 are 1, 2, 3, 6 and (1 + 2 + 3 + 6)

= (2 × 6). 28 is also a perfect number. Q
U
dTwo numbers are said to be co-prime if they do lTwo prime numbers are I
not have a common factor other than 1. always co-prime. C
K
lTwo co-primes need not

e.g., 6, 7 are co-primes, while 6 is not a prime be prime numbers.

number. TIP

dEvery even number greater than 4 can be expressed as the sum of two odd prime
numbers.

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e.g., (i) 6 = 3 + 3 Advance Tip
(ii) 8 = 3 + 5

Tests of divisibility of numbers : If a number is
dTest of divisibility by 2 : divisible by the two
co-prime numbers,
A number is divisible by 2, if its units digit is 0, 2, 4, 6 or 8. then it is divisible by
e,g., 32, 84, 240, 1396, 4258 etc. are divisible by 2. their product also.
dTest of divisibility by 3 :

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

e.g., Consider the number 372156.

Sum of its digits = (3 + 7 + 2 + 1 + 5 + 6) = 24, which is divisible by 3.

\372156 is divisible by 3.

dTest of divisibility by 4 :

A number is divisible by 4, if the number formed by its digits in tens and units place
is divisible by 4.

e.g., Consider the number 39256.

The number formed by tens and units digit is 56, which is divisible by 4.

\39256 is divisible by 4.

dTest of divisibility by 5 :

A number is divisible by 5, if its units digit is 0 or 5.

e.g., 15, 35, 80, 90, 1435 etc., are divisible by 5.

dTest of divisibility by 6 :

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

e.g., Consider the number 35484.

Its units digit is 4. So it is divisible by 2.

Sum of its digits = 3 + 5 + 4 + 8 + 4 = 24, which is divisible by 3.

Therefore, 35484 is divisible by both 2 and 3. And hence it is divisible by 6.

dTest of divisibility by 8 :

A number is divisible by 8, if the number formed by its digits in hundreds, tens and
units places is divisible by 8.

e.g., Consider the number 81424.

The number formed by hundreds, tens and units digits is 424 which is divisible by
8. So, 81424 is divisible by 8.

dTest of divisibility by 9 :

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

e.g., Consider the number 45207.

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Sum of its digits = 4 + 5 + 2 + 0 + 7 = 18, which is divisible by 9. Therefore,
45207 is divisible by 9.
dTest of divisibility by 10 :
A number is divisible by 10, if its units digit is zero.
e,g., 10, 30, 140, 480, 720, 2050, etc,. are divisible by 10.
dTest of divisibility by 11 :
A number is divisible by 11, if the difference of the sum of its digits in odd places
and sum of its digits in even places (starting from units place) is either 0 or a
multiple of 11.
e.g., Consider the number 530629.
Sum of its digits in odd places = 9 + 6 + 3 = 18
Sum of the digits in even places = 2 + 0 + 5 = 7
Difference of these sums = 18 – 7 = 11 which is divisible by 11.
Therefore, 530629 is divisible by 11.
dEvery composite number can be factorised into primes in only one way, except for
the order of primes. This property is known as unique factorization property.
dLeast Common Multiple (L.C.M.)
The Least Common Multiple (L.C.M.) of given numbers can be found by the
following methods.
(i) By Listing Multiples : List the multiples of the given numbers and then find the

least of the common multiples.
Multiples of 36 are 36, 72, 108, 144, 180, 216, ....
Multiples of 72 are 72, 144, 216, 288, 360, 432, .....
Observe that the common multiples are highlighted.
\The least of the common multiples is 72. Hence, L.C.M. of 36 and 72 is 72.
(ii)By Prime Factorisation Method (Division Method) : Divide the given numbers
by prime factors as shown below.
2 36, 72
2 18, 36
3 9, 18
3 3, 6

1, 2
The product of all the prime factors gives the L.C.M.
\L.C.M. = 2 × 2 × 3 × 3 × 2 = 72

L.C.M. of 36 and 72 = 72

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dHighest Common Factor (H.C.F.)
The highest common factor of the given numbers can be found by the following
methods.
(i) By Listing Factors : List out the factors of the given numbers and find the
highest of the common factors.
Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Observe that the common factors are highlighed.
Common factors of 24 and 36 = 1, 2, 3, 4, 6 and 12.

\The highest of the common factors is 12.
Hence, H.C.F. of 24 and 36 = 12.
(ii)Division Method : Divide the larger of the given numbers by the smaller one.
Subtract and then divide the smaller number by the remainder.
Continue until the remainder is 0. The last divisor is the required H.C.F. of the
numbers.

24) 36 ( 1
24
12) 24 (2
24
0

\H.C.F. of 24 and 36 = 12

(iii) By Prime Factorisation Method (Division Method) :

2 36 2 24 lThe H.C.F. of given numbers is not Q
greater than any of the given numbers. U
2 18 2 12 I
lThe H.C.F. of two co-primes is 1. C
39 26 K

33 33 lThe L.C.M. of given numbers is not less

11 than any of the given numbers. TIP

36 = 2 × 2 × 3 × 3 lThe L.C.M. of two co-primes is equal to

their product.

24 = 2 × 2 × 2 × 3 lThe H.C.F. of two given numbers is

The product of the common always a factor of their L.C.M.
factors = 2 × 2 × 3 = 12.

\H.C.F. of 36 and 24 is 12.

dProduct of two numbers = Product of their H.C.F. and L.C.M.

e.g., Consider the number 24 and 36.

L.C.M. of 24 and 36 = 72

H.C.F. of 24 and 36 = 12

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Product of numbers = 24 × 36 = 864
Product of H.C.F. and L.C.M. = 12 × 72 = 864

\Product of two numbers = Product of their H.C.F. and L.C.M.
dIf two numbers are separately divisible by a number, then their sum and difference

are also divisible by that number.

W arm Up

A. Choose the correct option from the four options given below. Mark your choice in the
answer sheet printed at the end of the questions.

1. Which of the following numbers is not exactly divisible by 5 ?

(a) 2105 (b) 5501 (c) 75500 (d) 55555

2. Which of the following numbers is not exactly divisible by 6 and 9 ?

(a) 432 (b) 1944 (c) 5508 (d) 2710

3. Which of the following is a prime number?

(a) 117 (b) 171 (c) 179 (d) 243

4. Which of the following is a composite number?

(a) 37 (b) 76 (c) 97 (d) 23

5. The HCF of 144 and 198 is :

(a) 9 (b) 12 (c) 6 (d) 18

6. The LCM of 24, 36 and 40 is :

(a) 4 (b) 90 (c) 360 (d) 720

7. The product of two numbers is 2160 and their HCF is 12. What is their LCM?

(a) 12 (b) 180 (c) 540 (d) 720

8. On simplifying 289 we get :
391
(a) 11 (b) 13 (c) 17 (d) 17
23 31 31 23

9. The prime factorisation of 70 is :

(a) 1 × 7 × 10 (b) 1 × 2 × 5 × 7 (c) 2 × 5 × 7 (d) 7 × 10

10. Which of the following pairs is of twin primes?

(a) (81, 83) (b) (53, 55) (c) (31, 37) (d) (5, 7)

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Practice Session

11. The numbers which are not multiples of 2 are called ___________ numbers.

(a) even (b) odd (c) prime (d) composite

12. The numbers (4, 8) are ___________ numbers.

(a) co-prime (b) prime

(c) both (a) and (b) (d) none of these

13. The two consecutive prime numbers with difference 2 are called :

(a) co-primes (b) twin primes (c) composite (d) even

14. The HCF of two numbers is 28 and their LCM is 336. If one number is 112, then
the other number is :

(a) 64 (b) 84 (c) 34 (d) 92

15. Two tankers contain 850 litres and 680 litres of kerosene oil respectively. Find the
maximum capacity of a container which can measure the kerosene oil of either
tanker in exact number of times :

(a) 170 litres (b) 87 litres (c) 34 litres (d) 10 litres

16. LCM of two co-prime numbers is their :

(a) Sum (b) Difference (c) Product (d) Quotient

17. Number of prime numbers from 1 to 50 are :

(a) 18 (b) 12 (c) 15 (d) 20

18. What least value should be given to * so that the number 653 * 47 is divisible by
11 ?

(a) 9 (b) 6 (c) 7 (d) 1

19. When a number is divided by 7, it has a remainder of 2. When it is divided by 6, it
has a remainder of 3. What is the number ?

(a) 16 (b) 38 (c) 44 (d) 9

20. The two numbers which have only 1 as their common factor are called :

(a) co-primes (b) twin primes (c) composite (d) even numbers

ADVANCELEVEL

21. Which of the following is smallest prime number ?

(a) 1 (b) 2 (c) 3 (d) 4

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22. The only prime number which is also even is :

(a) 1 (b) 2 (c) 4 (d) 6

23. The sum of two odd and one even numbers is :

(a) even (b) odd (c) prime (d) composite

24. The smallest composite number is :

(a) 1 (b) 2 (c) 3 (d) 4

25. Tell the maximum consecutive numbers less than 100 so that there is no prime
number between them :

(a) 5 (b) 6 (c) 7 (d) 8

26. If a number is divisible by 2 and 3 both then it is divisible by :

(a) 5 (b) 6 (c) 8 (d) 10

27. Which of the following number is divisible by 3 ?

(a) 121 (b) 123 (c) 124 (d) 122

28. A number is divisible by 4 if its :

(a) last digit is 4 (b) last digit is 0

(c) last two digits are divisible by 4 (d) last digit is 8

29. Common factors of 15 and 25 are :

(a) 15 (b) 25 (c) 5 (d) 75

30. If a number is divisible by two co-prime numbers then it is divisible by their :

(a) sum also (b) difference also (c) product also (d) quotient also

1. a b c d 2. a b c d 3. a b c d 4. a bc d
5. a b c d 6. a b c d 7. a b c d 8. a bc d
9. a b c d 10. a b c d 11. a b c d 12. a bc d
13. a b c d 14. a b c d 15. a b c d 16. a bc d
17. a b c d 18. a b c d 19. a b c d 20. a bc d
21. a b c d 22. a b c d 23. a b c d 24. a bc d
25. a b c d 26. a b c d 27. a b c d 28. a bc d
29. a b c d 30. a b c d

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