CBSE Class 10 Mathematics Real Numbers Worksheet Set H

Question. The \( LCM \) of two numbers is 198 and their product is 1188. Find their \( HCF \)
(a) 18
(b) 6
(c) 66
(d) None of the options
Answer: (b) 6

Question. If two positive integers and are expressible in terms of primes as \( a = p^6 q^3 \) and \( b = p^4 q^8 \), then which of the following is true?
(a) \( HCF = p^5q^2 \times LCM \)
(b) \( LCM = p^2q^5 \times HCF \)
(c) \( LCM = p^5q^2 \times HCF \)
(d) \( HCF = p^2q^5 \times LCM \)
Answer: (b) \( LCM = p^2q^5 \times HCF \)

Question. The values of x and y in the given figure are:
[Figure: y branches into 4 and x; x branches into 3 and 7]
(a) \( x = 10; y = 14 \)
(b) \( x = 21; y = 84 \)
(c) \( x = 21; y = 25 \)
(d) \( x = 10; y = 40 \)
Answer: (b) \( x = 21; y = 84 \)

Question. The \( HCF \) and \( LCM \) of two numbers are 33 and 264 respectively. When the first number is completely divided by 2 the quotient is 33. The other number is:
(a) 66
(b) 130
(c) 132
(d) 196
Answer: (c) 132

Question. What will be the least possible number of the planks, if three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length?
(a) 5
(b) 6
(c) 7
(d) None of the options
Answer: (c) 7

Question. What is the greatest possible speed at which a man can walk 52 km and 91 km in an exact number of minutes?
(a) 17 m/min
(b) 7 m/min
(c) 13 m/min
(d) 26 m/min
Answer: (c) 13 m/min

Question. If \( A = 2n + 13 \), \( B = n + 7 \), where n is a natural number then \( HCF \) of A and B is:
(a) 2
(b) 1
(c) 3
(d) 4
Answer: (b) 1

Question. Pairs of natural numbers whose least common multiple is 78 and the greatest common divisor is 13 are:
(a) 58 and 13 or 16 and 29
(b) 68 and 23 or 36 and 49
(c) 18 and 73 or 56 and 93
(d) 78 and 13 or 26 and 39
Answer: (d) 78 and 13 or 26 and 39

Question. Two natural numbers whose sum is 85 and the least common multiple is 102 are:
(a) 30 and 55
(b) 17 and 68
(c) 35 and 55
(d) 51 and 34
Answer: (d) 51 and 34

Question. 4 Bells toll together at 9.00 am. They toll after 7, 8, 11 and 12 seconds respectively. How many times will they toll together again in the next 3 hours?
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (c) 5

Question. A forester wants to plant 66 apple trees, 88 banana trees and 110 mango trees in equal rows (in terms of number of trees). Also he wants to make distinct rows of trees (i.e., only one type of trees in one row). The number of minimum rows required are
(a) 2
(b) 3
(c) 10
(d) 12
Answer: (d) 12

Question. The largest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is
(a) 13
(b) 65
(c) 875
(d) 1750
Answer: (a) 13

Question. If the \( HCF \) of 65 and 117 is expressible in the form \( 65m - 117 \), then the value of m is
(a) 4
(b) 2
(c) 1
(d) 3
Answer: (b) 2

Question. If two positive integers a and b are written as \( a = x^3y^2 \) and \( b = xy^3 \); x, y are prime numbers, then \( HCF (a, b) \) is
(a) \( xy \)
(b) \( xy^2 \)
(c) \( x^3y^3 \)
(d) \( x^2y^2 \)
Answer: (b) \( xy^2 \)

Question. If two positive integers p and q can be expressed as \( p = ab^2 \) and \( q = a^3b \); a, b being prime numbers, then \( LCM (p, q) \) is
(a) \( ab \)
(b) \( a^2b^2 \)
(c) \( a^3b^2 \)
(d) \( a^3b^3 \)
Answer: (c) \( a^3b^2 \)

Direction: In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.

Question. Assertion: The \( H.C.F. \) of two numbers is 16 and their product is 3072. Then their \( L.C.M. = 162 \).
Reason: If a and b are two positive integers, then \( H.C.F. \times L.C.M. = a \times b \).
(a) A
(b) B
(c) C
(d) D
Answer: (d) Assertion (A) is false but Reason (R) is true.

Question. Assertion: Denominator of 34.12345. When expressed in the form p/q, \( q \neq 0 \), is of the form \( 2^m \times 5^n \), where m and n are non-negative integers.
Reason: 34.12345 is a terminating decimal fraction.
(a) A
(b) B
(c) C
(d) D
Answer: (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Question. Assertion: A number N when divided by 15 gives the remainder 2. Then the remainder is same when N is divided by 5.
Reason: \( \sqrt{3} \) is an irrational number.
(a) A
(b) B
(c) C
(d) D
Answer: (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

Question. Assertion: 2 is an example of a rational number.
Reason: The square roots of all positive integers are irrational numbers.
(a) A
(b) B
(c) C
(d) D
Answer: (c) Assertion (A) is true but Reason (R) is false.

Question. Assertion: For any two positive integers p and q, \( HCF (p, q) \times LCM (p, q) = p \times q \)
Reason: If the \( HCF \) of two numbers is 5 and their product is 150, then their \( LCM \) is 40.
(a) A
(b) B
(c) C
(d) D
Answer: (c) Assertion (A) is true but Reason (R) is false.

Direction: In the following questions, short answer of 2 marks each

Question. Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11, and 15 respectively.
Answer: 17

Question. Express 98 as a product of its primes.
Answer: \( 2 \times 7^2 \)

Question. Zoe and Sam are racing on a circular track. If Zoe takes 48 minutes and Sam takes 80 minutes to complete the round. If they both start at the same point at the same time and go in same direction, after how many minutes will they meet again at the start point?
Answer: 240 mins

Question. Mitchell and Courtney are racing on a circular track. If Mitchell takes 36 minutes and Courtney takes 24 minutes to complete the round. If they both start at the same point at the same time and go in same direction, then they will meet again at the start point after how many minutes.
Answer: 72 mins

Question. In a seminar, the number of participants in German, English and French are 130, 130 and 286 respectively. Find the numbers of rooms required to house them if in each room, the same number of participants are to be accommodated and all of them must belong to the same language
Answer: 26

Question. If the \( HCF \) of 408 and 1032 is expressible in the form \( 1032 \times 2 + 408 \times p \), then find the value of p.
Answer: -5

Question. \( HCF \) and \( LCM \) of two numbers is 9 and 459 respectively. If one of the numbers is 27, find the other number
Answer: 153

Question. Find \( HCF \) and \( LCM \) of 13 and 17 by prime factorisation method
Answer: \( HCF=1 ; LCM= 221 \)

Question. Find \( LCM \) of numbers whose prime factorisation are expressible as \( 3 \times 5^2 \) and \( 3^2 \times 7^2 \) .
Answer: \( 3^2 \times 5^2 \times 7^2 \)

Question. Find the \( LCM \) of 96 and 360 by using fundamental theorem of arithmetic.
Answer: 1440

Direction: In the following questions, short answer of 3 marks each

Question. Karan has 180 blue marbles and 150 red marbles. He wants to pack them into packets containing equal number of marbles of the same colour. What is the maximum number of marbles that each packet can hold?
Answer: 30

Question. Find the largest number that will divide 382 and 710 and leaves a remainder 13 in each case.
Answer: 41

Question. What is the largest number that divides 437, 732, and 1263 leaving remainder of 24 in each case?
Answer: 59

Question. What is the largest number that divides 967 and 1767 leaving remainders of 71 and 103 respectively?
Answer: 128

Question. What is the largest number that divides 170, 220, and 420 leaving remainder 8, 4 and 15 respectively?
Answer: 27

Question. Find the \( LCM \) and \( HCF \) of the following :
\( 2^5 \times 5^4 \times 7^2 \times 13^6 \) and \( 2^3 \times 5^6 \times 7 \times 17^3 \).
Answer: \( LCM = 2^5 \times 5^6 \times 7^2 \times 13^6 \times 17^3 \); \( HCF = 2^3 \times 5^4 \times 7 \)

Question. Prove that \( 2-3\sqrt{5} \) is an irrational number
Answer: Irrational

Question. The \( LCM \) of two numbers is 14 times their \( HCF \). The sum of \( LCM \) and \( HCF \) is 600. If one number is 280, then find the other number.
Answer: 320

Question. An army contingent of 1000 members is to march behind an army band of 56 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
Answer: 8

Question. Three bells toll at intervals of 12 minutes, 15 minutes and 18 minutes respectively. If they start tolling together, after what time will they next toll together?
Answer: 90 mins

Long Answer Type Question

Question. Find \( HCF \) of 378, 180 and 420 by prime factorisation method. Is \( HCF \times LCM \) of three numbers equal to the product of the three numbers?
Answer: 6 ; YES

Question. The Muscle Gym has bought 63 treadmills and 108 elliptical machines. The gym divides them into several identical sets of treadmills and elliptical machines for its branches located throughout the city, with no exercise equipment left over. What is the greatest number of branches the gym can have in the city?
Answer: 9

Question. Katya has 49 paintings and 35 medals. She wants to display them in groups throughout her house, each with the same combination of paintings and medals, with none left over. What is the greatest number of groups Katya can display?
Answer: 7

Question. Anish goes fishing every 5th day and Balaji goes fishing every 7th day. If Anish and Balaji both went fishing today, how many days until they will go fishing on the same day again?
Answer: 35

Question. Tamanna is arranging black marbles in groups of 13 and purple marbles in groups of 25. If she has the same number of black and purple marbles, what is the smallest number of marbles of each colour that she could have?
Answer: 325

CASE STUDY QUESTION

The department of Computer Science and Technology is conducting an International Seminar. In the seminar, the number of participants in Mathematics, Science and Computer Science are 60, 84 and 108 respectively. The coordinator has made the arrangement such that in each room, the same number of participants are to be seated and all of them being in the same subject. Also, they allotted the separate room for all the official other than participants.

Question. Find the total number of participants.
(a) 60
(b) 84
(c) 108
(d) None of the options
Answer: 252

Question. Find the \( LCM \) of 60, 84 and 108.
(a) 12
(b) 504
(c) 544320
(d) 3780
Answer: 3780

Question. Find the \( HCF \) of 60, 84 and 108.
Answer: 12

Question. Find the minimum number of rooms required, if in each room, the same number of participants are to be seated and all of them being in the same subject.
(a) 12
(b) 20
(c) 21
(d) None of the options
Answer: 21

Question. Based on the above (iv) conditions, find the minimum number of rooms required for all the participants and officials.
(a) 12
(b) 20
(c) 21
(d) None of the options
Answer: 22q