CBSE Class 10 Mathematics Real Numbers VBQs

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

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

Question. If two positive integers a and b are written as a = x³y² and b = xy³; x, y are prime numbers, then HCF (a, b) is
(a) xy
(b) xy²
(c) x³y³
(d) x²y²
Answer : B

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

Question. If two positive integers p and q can be expressed as p = ab² and q = a3b; a, b being prime numbers, then LCM (p, q) is
(a) ab
(b) a²b²
(c) a³b²
(d) a³b³
Answer : C

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

Question. Express 98 as a product of its primes.
Answer : 2 X 7²

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

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

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

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

Question. Find LCM of numbers whose prime factorisation are expressible as 3 × 5² and 3² × 7².
Answer : 3² x 5² x 7²

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. 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

Question. Prove that 2-3√5 is an irrational number
Answer : Irrational

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

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

Question. Find the LCM and HCF of the following : 2⁵ × 5⁴ × 7² × 13⁶ and 2³ × 5⁶ × 7 × 17³.
Answer : LCM=2⁵ x 5⁶ x 7² x 13⁶ x 17³ ; HCF = 2³ x 5⁴ x 7

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

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. 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. 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

Question. The product of two numbers is 228096 and their LCM is 66. Find their HCF.
Answer : 36

Question. If two positive integers a, b are written as a = 𝑥𝑦² and b = 𝑥³𝑦, where x, y are prime numbers, then find LCM (a, b).
Answer : X³Y²

Question. The difference of the irrational numbers 5 + √2 and 5 - √2?
Answer : 2√2

Short Answer type Questions

Question. Two brands of chocolates are available in packs of 24 and 15 respectively. If I need to buy an equal number of chocolates of both kinds, what is the least number of boxes of each kind I would need to buy?
Answer : 5 of 1^st kind, 8 of 2^nd kind

Question. Find the LCM and HCF of the following pairs of positive integers by applying the prime factorization method.
a) 225, 240 b) 52 ,63 ,162
Answer : a) HCF (225, 240 ) = 15 LCM (225, 240) = 600
b) HCF (52, 6, 162) = 1 LCM (52, 63, 162) = 29484

Question. The LCM of two numbers is 64699, their HCF is 97 and one of the numbers is 2231.Find the other.
Answer : 2813

Question. If the sum of LCM and HCF of two numbers is 1260 and their LCM is 900 more than their HCF then, find the product of two numbers.
Answer : 194400

Question. Find HCF and LCM of 135 and 225 and verify the that HCF x LCM = Product of the two given numbers.
Answer : LCM (135, 225) = 675, HCF (135, 225) = 45. Verification by showing LHS = RHS i.e., 135 x 225 = 675 x 45

Question. Find HCF and LCM of 867 and 255 and verify the that HCF x LCM = Product of the two given numbers
Answer : LCM (867, 255) = 4335, HCF (867, 255) = 51. Verification by showing LHS = RHS i.e., 867 x 255 = 4335 x 51

Question. What is the LCM of smallest prime number and smallest composite number?
Answer : 4

Question. Is (√2 + √3 )² and (2- √2) (2 + √2) irrational? Justify your answer.
Answer : (√2 + √3 )² is irrational as the result is 5 + √6 , which is irrational.
(2- √2 ) (2 + √2 ) is rational as the result is 2, which is rational.

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 theofficial other than participants.

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

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 these
Answer : 21

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

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

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