Class 10 Maths Extra Questions Chapter 1 Real Numbers

VERY SHORT ANSWER TYPE QUESTIONS
1. A number N when divided by 16 gives the remainder 5 ______ is the remainder
when the same number is divided by 8.
2. HCF of 33 × 54 and 34 × 52 is ________ .
3. If a = xy2 and b = x3y5 where x and y are prime numbers then LCM of (a, b) is
_____ .
4. In the given factor tree find x and y

y

2 x

5 7

5. If n is a natural number, then 252n – 92n is always divisible by :
(i) 16 (ii) 34
(iii) both 16 or 34 (iv) None of these
327
6. The decimal expansion of the rational number will terminate after
23  5
(a) One decimal place (b) Two decimal place
(c) Three decimal place (d) More than three decimal place
7. Which of the following rational numbers have terminating decimal?
18 5 2 7
(i) (ii) (iii) (iv)
225 18 21 250
(a) (i) and (ii) (b) (ii) and (iii)
(c) (i) and (iii) (d) (i) and (iv)
8. Euclid’s division Lemma states that for two positive integers a and b, there
exist unique integers q and r such that a = bq + r, where r must satisfy.
(a) 1 < r < b (b) 0 < r  b
(c) 0  r < b (d) 0 < r < b
n n n
9. p = (a × 5) For p to end with the digit zero a = _____ for natural number n.
(a) any natural number (b) even number
(b) odd number (d) none of these
2 Mathematics-X

10. HCF is always
(a) multiple of LCM (b) Factor of LCM
(c) divisible by LCM (d) (a) and (c) both
11. All decimal numbers are
(a) rational number (b) irrational numbers
(c) real numbers (d) integers
12. Which of these numbers always end with the digits 6.
(a) 4n (b) 2n (c) 6n (d) 8n
13. Write the prime factof of 2 × 7 × 11 × 13 × 17 + 21
14. Write the form in which every odd integer can be written taking t as variable.
15. What would be the value of n for which n2–1 is divisible by 8.
16. What can you say about the product of a non-zero rational and irrational number?
13497
17. After how many places the decimal expansion of will terminate?
1250
18. Find the least number which is divisible by all numbers from 1 to 10 (both
inclusive).
19. The numbers 525 and 3000 are divisible by 3, 5, 15, 25 and 75. What is the
HCF of 525 and 3000?
20. What is x : y in the factor-tree?
x

2 210

2 105

3 35

5 y

SHORT ANSWER TYPE QUESTIONS-I
21. If n is an odd integer then show that n2 – 1 is divisible by 8.
22. Use Euclid’s division algorithm to find the HCF of 16 and 28.
23. Show that 12n cannot end with the digit 0 or 5 for any natural number n.
(NCERT Exemplar)
Mathematics-X 3

395
24. Without actually performing the long division, find if will have
10500
terminating or non terminating (repeating decimal expansion.)
25. A rational number in its decimal expansion is 327. 7081. What can you say
p
about the prime factors of q, when this number is expressed in the form of ?
q
Give reasons.
26. What is the smallest number by which 5 – 2 is to be multiplied to make it
a rational number? Also find the number so obtained?
27. Find one rational and one irrational no between 3 and 5.
28. If HCF of 144 and 180 is expressed in the form 13m – 3, find the value of m.
(CBSE 2014)
29. Find the value of : (–1)n + (–1)2n + (–1)2n + 1 + (–1)4n+2, where n is any positive
odd integer. (CBSE 2016)
30. Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is
some integer. (CBSE 2012)
31. Two tankers contain 850 litres and 680 litres of petrol respectively. Find the
maximum capacity of a container which can measure the petrol of either tanker
in exact number of times. (CBSE 2016)

SHORT ANSWER TYPE QUESTIONS-II
32. Show that the cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3
for some integer m.
33. If 7560 = 23 × 3p × q × 7, find p and q.
105
34. If is a terminating decimal, what are the least possible values
2  5  3 p  7q
3 2

of p and q.
35. Prove that 3 + 5 is irrational

3
36. Prove that 5 – 3 is an irrational number..
7

4 Mathematics-X

1
37. Prove that is an irrational number..
2– 5

38. Find HCF and LCM of 56 and 112 by prime factorization method.
39. Explain why:
(i) 7 × 11 × 13 × 15 + 15 is a composite number
(ii) 11 × 13 × 17 + 17 is a composite number.
(iii) 1 × 2 × 3 × 5 × 7 + 3 × 7 is a composite number.
40. On a morning walk, three persons steps off together and their steps measure 40
cm, 42 cm, and 45 cm respectively. What is the minimum distance each should
walk, so that each can cover the same distance in complete steps?
(NCERT Exemplar)
41. During a sale, colour pencils were being sold in the pack of 24 each and crayons
in the pack of 32 each. If you want full packs of both and the same number of
pencils and crayons, how many packets of each would you need to buy?
(CBSE : 2017)
42. Find the largest number that divides 31 and 99 leaving remainder 5 and 8
respectively.
43. The HCF of 65 and 117 is expressible in the form 65 m – 117. Find the value of
m. Also find the LCM of 65 and 117 using prime factorisation method.
44. Using Euclid’s division algorithm, find the largest number that divides 1251,
9377 and 15628 leaving remainder 1, 2 and 3 respectively.
(NCERT Exemplar)
45. Show that square of any odd integer is of the form 4m + 1, for some integer m.
46. Find the HCF of 180, 252 and 324 by Euclid’s Division algorithm.
47. Find the greatest number of six digits exactly divisible by 18, 24 and 36.
48. Three bells ring at intervals of 9, 12, 15 minutes respectively. If they start
ringing together at a time, after how much time will they next ring together?
49. Show that either the number n, n + 2 and n + 4 are three consecutive odd
numbers or 3 even numbers.

Mathematics-X 5

50. Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of
two given number. (CBSE 2018)

LONG ANSWER TYPE QUESTIONS
51. Find the HCF of 56, 96, 324 by Euclid’s algorithm.
52. Show that any positive odd integer is of the form 6q + 1, 6q + 3 or 6q + 5,
where q is some integer.
53. Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for
some integer, q.
54. Prove that the product of three consecutive positive integers is divisible by 6.
55. For any positive integer n, prove that n3–n is divisible by 6.
(NCERT Exemplar)
56. Show that one and only one of n, n + 2, n + 4 is divisible by 3.
57. Aakriti decided to distribute milk in an orphanage on her birthday. The supplier
brought two milk containers which contain 398 l and 436 l of milk. The milk is
to be transferred to another containers so that 7 l and 11 l of milk is left in both
the containers respectively. What will be the maximum capacity of the
drum?
58. Find the smallest number, which when increased by 17, is exactly divisible by
both 520 and 468.
59. A street shopkeeper prepares 396 Gulab jamuns and 342 ras-gullas. He packs
them, in combination. Each containter consists of either gulab jamuns or ras-
gullab but have equal number of pieces.
Find the number of pieces he should put in each box so that number of boxes
are least. How many boxes will be packed in all. (CBSE 2016)
60. Show that the square of any positive integer cannot be of the form 5q + 2 or
5q + 3 for integer q.
61. Express the HCF of numbers 72 and 124 as a linear combination of 72 and 124.

62. Show that there is no positive integer n for which n 1 + n  1 is rational.
63. Find the number nearest to 110000 but greater than 1 lakh, which is exactly
divisible by 8, 15, 21.
64. In a seminar, the no. of participants in Hindi, English and Mathematics are 60,
84 and 108 respectively. Find the minimum number of rooms required if in

6 Mathematics-X

each room the some the same number of participants are to be seated and all of
the them being of the the same subject. (HOTS)
65. State fundamental theorem of Arithmetic. Is it possible that HCF and LCM of
two numbers be 24 and 540 respectively. Justify your answer.
66. Find the smallest number which when increased by 20 is exactly divisible by
90 and 144. If we express the smallest number, the LCM, in the form 144 m +
90n, then what are the values of m and n? Is LCM, a multiple of 144?
67. If the HCF of 1032 and 408 is expressible in the form 1032 p – 408×5, find p.
68. The LCM of two numbers is 14 times their HCF. The sum of LCM and HCF is
600. If one of the number is 280. Find the other number.

ANSWERS AND HINTS

1. 5 2. 33 × 52
3. x3 × y5 4. x = 35, y = 70
5. (iii) 252n – 92n is of the form a2n – b2n which is divisible by both a – b and
a + b so, by both 25 + 9 = 34 and 25 – 9 = 16.
6. (c) three decimal place 7. (d) (i) and (iv)
8. (c) 0  r < b 9. (b) even number
10. (b) Factor of LCM 11. (c) real numbers
12. (c) 6n 13. 7
14. 2t + 1 15. An odd integer
16. Irrational 17. 4
18. 2520 19. 75
20. 60 : 1
21. Any +ve odd integer is of the form 4q + 1 or 4q + 3 for some integer q so if
n = 4q + 1.
n2 – 1 = (4q + 1)2–1 = 16q2 + 8q = 8q(2q + 1)  n2 – 1 is divisible by 8.
If n = 4q + 3
n2 – 1 = (4q + 3)2 – 1 = 16q2 + 24q + 8 = 8 (2q2 + 3q + 1)  n2 – 1 is divisible
by 8.

Mathematics-X 7

22. 4
23. As 12 has factors 2, 2, 3 it doesnot has 5 as its factor so 12 n will never end
with 0 or 5.
24. Non-terminating repeating.
25. Denominator is the multiple of 2’s and 5’s.
26. 5 2 , 3
28. By Euclid’s division lemma
180 = 144 × 1 + 36
144 = 36 × 4 + 0
HCF of 180 and 144 is 36.
13m – 3 = 36
13 m = 39
m= 3
29. Given that n is a positive odd integer
 2n and 4n + 2 are even positive integers and n and 2n + 1 are odd positive
integers.
 (–1)n = – 1, (–1)2n = + 1, (–1)2n + 1 = – 1, (–1)2n + 2 = + 1
 (–1)n + (–1)2n + (–1)2n + 1 + (–1)4n + 2 = – 1 + 1 – 1 + 1 = 0
30. By applying Euclid division algorithm to a and b such that a = 4q + r, where
b = 4, Now r = 0, 1, 2, 3.
where, r = 0, a = 4q which is even number.
where, r = 1, a = 4q + 1 an odd number.
where, r = 2, a = 4q + 2 =2 (2q + 1), an even number.
where, r = 3, a = 4q + 3 an odd number.
31. HCF of 850 and 680 is 2 × 5 × 17 = 170 litres.
32. Let n be any psoitve integer. Then it is of the form 4q, 4q + 1, 4q + 2 and
4q + 3.
When n = 4q, n3 = 64q3 = 4 (16q3) = 4 m, where m = 16q3
When n = 4q + 1, n3 = (4q + 1)3 = 64q3 + 48q2 + 12q + 1

8 Mathematics-X

= 4(16q3 + 12q2 + 3q) + 1 = 4 m + 1.
where m = 16q3 + 12q2 + 3q
Similarly discuss for n = 4q + 2 and 4q + 3.
33. p = 3 and q = 5
34. 0 and 1
35. Prove that 3 and 5 is irrational number separately..
3
36. 5 is rational no. and 3 is an irrational number. Difference of a rational
7
number and irrational number is an irrational number.
38. HCF : 56, LCM : 112
39. (1) 15 × (7 × 11 × 13 + 1) as it has more than two factors so it is composite no.
40. LCM of 40, 42, 45 = 2520
Minimum distance each should walk 2520 cm.
41. LCM of 24 and 32 is 96
96
96 crayons or = 3 packs of crayons
32
96
96 pencils or = 4 packs of pencils.
24
42. Given number = 31 and 99
31 – 5 = 26 and 99 – 8 = 91
Prime factors of 26 = 2 × 13
91 = 7 × 13
HCF of (26, 91) = 13.
 13 is the largest number which divides 31 and 99 leaving remainder 5
and 8 respectively.
43. HCF of 117 and 65 by Euclid division algorithm.
117 = 65 × 1 + 52
65 = 52 × 1 = 52
52 = 13 × 4 + 0
HCF (117, 52) = 13.
Given that 65 m – 117 = 13  65 m = 130  m = 2.
LCM (65, 117) = 13 × 32 × 5 = 585
Mathematics-X 9

44. 1251 – 1 =1250, 9377 – 2 = 9375, 15628 – 3 = 15625
HCF of (15625, 9375) = 3125
HCF of (3125, 1250) = 625
 HCF of (1250, 9375, 15625) = 625
45. By Euclid’s division algorithm, we have a = bq + r, where 0  r < 4. On
putting b = 4 we get a = 4q + r where, r = 0, 1, 2, 3.
If r = 0, a = 4q which is even
If r = 1, a = 4q + 1 not divisible by 2
If r = 2, a = 4q + 2 = 2(2q + 1) which is even
If r = 3, a = 4q + 3 not divisible by 2.
So, for any +ve integer q, 4q + 1 and 4q + 3 are odd integers.
How, a2 = (4q + 1)2 = 16q2 + 1 + 8q = 4(4q2 + 2q) + 1 = 4m + 1
where m = 4q2 + 2q similarly for 4q + 3.
46. HCF (324, 252, 180) = 36
47. LCM of (18, 24, 36) = 72.
Greatest six digit number = 999999

72 999999 13888
– 72
279
– 216
639 Require six digit number
– 576
999999
639 – 63
– 576
639 999936
576
63
48. LCM of (9, 12, 15) = 180 minutes.
49. Let the number divisible by 3 is of the form 3k + r, r = 0, 1, 2
a = 3k, 3k + 1 or 3k + 2
(i) When a = 3k
n = 3k  n is divisible by 3.
n + 2 = 3k + 2  n + 2 is not divisible by 3.
n + 4 = 3k + 4 = 3k + 3 + 1 = 3(k + 1) + 1  n + 4 is not
divisible by 3.
So, only one out of n, n + 2 and n + 4 is divisible by 3.
10 Mathematics-X

(ii) When a = 3k + 1
n = 3k + 1  n is not divisible by 3.
n + 2 = 3k + 1 + 2 = 3k + 3 = 3(k + 1)
 n + 2 is divisible by 3.
n + 4 = 3k + 1 + 4 = 3k + 5 = 3(k + 1) + 2
 n + 2 is not divisible by 3.
So, only one out of n, n + 2 and n + 4 is divisible by 3.
Similarly do for a = 3k + 2.
50. HCF (404, 96) = 4
LCM (404, 96) = 9696
HCF × LCM = 38, 784
Also, 404 × 96 = 38,784
51. 4
52. Let a be +ve odd integer.
 a = 6q + r where r = 0, 1, 2, 3, 4, 5
If, a = 6q + 0 = 2(3q) is an even integer so not possible
If, a = 6q + 1 is an odd integer
If, a = 6q + 2 = 2(3q + 1) is an even integer so not possible
If, a = 6q + 3 is an odd integer
If, a = 6q + 4 = 2(3q + 2) is an even integer so not possible
If, a = 6q + 5 is an odd integer.
54. Let the three consecutive integers be a, a+1, a + 2,
Case I : If a is even,
 a + 2 is the also even
a(a + 2) is divisible by 2
a(a + 2) (a + 1) is also divisible by 2
Now a, a + 1, a + 2 are three consecutive numbers
 a (a + 1) (a + 2) is a multiple by 3
 a (a + 1) (a + 2) is divisible by 3
as it is divisible by 2 and 3 hence divisible by 6.
Case II : If a is odd
 a + 1 is even

Mathematics-X 11

 a + 1 is divisible by 2
 a(a + 1) (a + 2) is also divisible by 2
Again a, a + 1, a + 2 are three consecutive numbers
 a (a + 1) (a + 2) is a multiple by 3
 a (a + 1) (a + 2) is divisible by 3
as it is divisible by 2 and 3 hence divisible by 6.
55. n3 – n = n(n2 – 1) = n (n – 1) (n + 1)
= (n – 1) (n) (n + 1)
= Product of three consecutive +ve integers
Now to show that produce of three consecutive +ve integers is divisible by 6.
Any +ve integer a is of the form 3q, 3q + 1 or 3q + 2 for some integer q.
Let a, a + 1, a + 2 be any three consecutive integers.
Case I : a = 3q
(3q) (3q + 1) (3q + 2) = 3q (2m) [as (3q + 1) and (3q + 2) are consecutive
integers so their product is also even]
= 6q m
which is divisible by 6.
Case II : If a = 3q + 1
a (a + 1) (a + 2) = (3q + 1) (3q + 2) (3q + 3)
= 2m3(q + 1) (as (3q + 1) (3q + 2) = 2m)
= 6 m (q + 1)
which is divisible by 6.
Case III : If a = 3q + 2
a (a + 1) (a + 2) = (3q + 2) (3q + 3) (3q + 4)
= (3q + 2) 3(q + 1) (3q + 4)
= 6m
which is divisible by 6.
57. 17
58. 4663
LCM of (468, 520) = 4680
 Required no. = 4680 – 17 = 4663
59. HCF (396, 342) = 18
396  342 738
No. of boxes = = = 41
18 18
12 Mathematics-X

61. HCF (124, 72) = 4
4 = 124 × 7 + 72 × (– 12), x = 7, y = – 12
p
62. Let n 1  n 1  (1)q  0, p, q, co-prime.
q

q 1 n 1 – n 1
= 
p n 1  n 1 n 1  n 1

q n 1 – n 1
=
p 2
2q 2q
n 1  n 1   or n 1  n 1  …(2)
p p

p 2 q p 2  2q 2
Adding (1) & (2) we get 2 n  1    …(3)
q p pq
2 2
Subtracting (1) & (2) we get 2 n  1  p  2q …(4)
pq

From (3) & (4) we get n  1  n  1 are rational numbers.
But n  1  n  1 is an irrational number..

 These exist no positive integer n, for which n  1  n  1 is rational.
63. 109200
64. HCF of 60, 84 and 108 is 22 × 3 = 12 = No. of participants in each row.
Total number of participants
No. of rooms required =
12
60  84  108
= = 21 rooms
12
65. HCF = 24, LCM = 540
LCM 540
= = 22.5, not an integer.
HCF 24
Hence two numbers cannot have HCF and LCM as 24 and 540 respectively.

Mathematics-X 13

66. [The LCM of (90, 144) – 20] = Required No.
 Required No. = 700
If 720 = 144m + 90n then m = 5, n = 0, Yes
67. p = 2
68. HCF = 40, LCM = 560
 Other No. = 80.

14 Mathematics-X

PRACTICE-TEST
Real Number
Time : 1 Hr. M.M. : 20
SECTION A
1. Check whether 17 × 19 × 21 × 23 + 7 is a composite number. 1
2. In Euclid’s Division Lemma, when a = bq + r where a, b are positive integers
then what values r can take? 1
3. HCF of x4y5 and x8y3 . 1
4. LCM of 14 and 122 . 1
SECTION B
5. Show that 9 can never ends with unit digit zero.
n
2
805
6. Without actual division find the type of decimal expansion of , if
10500
terminating, after how many places. 2
7. Show that the square of any odd integer is of the form 4m + 1, for some integer
m.
2

SECTION C
1
8. Prove that is an irrational number.. 3
3– 2 5

9. Find the HCF of 36, 96 and 120 by Euclid’s Lemma. 3

SECTION D
10. Once a sports goods retailer organized a campaign “Run to remember” to spread
awareness about benefits of walking. In that Soham and Baani participated.
There was a circular path around a sports field. Soham took 12 minutes to
drive one round of the field, while Baani took 18 minutes for the same. Suppose
they started at the same point and at the same time and went in the same direction.
After how many minutes have they met again at the starting point? 4

Mathematics-X 15