CBSE Class 10 Maths HOTs Real Numbers Set 08

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: (d) None of the options
(Correct answer is 22: \( \frac{42}{7} + \frac{49}{7} + \frac{63}{7} = 6 + 7 + 9 = 22 \))

Question. What is the greatest possible speed at which a man can walk 52 m and 91 m 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. HCF of \( (2^3 \times 3^3 \times 5) \), \( (2^2 \times 3^2 \times 5^2) \) and \( (2^5 \times 3 \times 5^3 \times 7) \) is
(a) 30
(b) 60
(c) 105
(d) 210
Answer: (b) 60

Question. If x and y are positive integers such that \( x = a^2b^3 \) and \( y = a^3b^2 \), where a, b are prime numbers, then LCM of (x, y) =
(a) \( a^2b^3 \)
(b) \( a^3b^2 \)
(c) \( a^3b^3 \)
(d) \( a^2b^2 \)
Answer: (c) \( a^3b^3 \)

Question. If LCM of 24 and 48 is \( 10m + 8 \) then value of m is
(a) 4
(b) 8
(c) 2
(d) 1
Answer: (a) 4

Question. The exponent of 2 in prime factorisation of 288 is
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (d) 5

Question. Two equilateral triangles have the sides of lengths 34 cm and 85 cm respectively. Find the greatest length of tape that can measure the sides of both of them exactly.
Answer: HCF(34, 85) = 17 cm.

Question. There is a circular path around a sports field. Kamal takes 32 minutes to drive one round of the field while Indu takes 24 minutes for the same. After how many minutes they meet again at the starting point?
Answer: LCM(32, 24) = 96 minutes.

Question. P is LCM of 2, 4, 6, 8, 10; Q is LCM of 1, 3, 5, 7, 9 and L is LCM of P and Q. Evaluate L – 21P.
Answer: P = 120, Q = 315. L = LCM(120, 315) = 2520. \( L - 21P = 2520 - 21(120) = 2520 - 2520 = 0 \).

Question. The largest number which divides 71 and 126 leaving remainder 6 and 9 respectively is
(a) 65
(b) 875
(c) 13
(d) 1750
Answer: (c) 13

Question. The smallest number, which when increased by 14 is exactly divisible by 165 and 770, is
(a) 2297
(b) 2310
(c) 2296
(d) 2295
Answer: (c) 2296

Question. The LCM and HCF of two non-zero positive numbers are equal, then the numbers must be
(a) composite
(b) prime
(c) co-prime
(d) equal
Answer: (d) equal

Question. The least number which when divided by 18, 24, 30 and 42 will leave same remainder 1, would be
(a) 2520
(b) 2519
(c) 2521
(d) 2522
Answer: (c) 2521

Question. If x and y are coprime then \( x^3 \) and \( y^3 \) are
(a) even
(b) odd
(c) co-prime
(d) not coprime
Answer: (c) co-prime

Question. In a school there are two sections, section A and section B of class X. There are 45 students in section A and 36 students in section B. The minimum numbers of books required for their class library so that they can be distributed equally among the students of section A or section B are
(a) 280
(b) 180
(c) 90
(d) 120
Answer: (b) 180

Question. HCF of 24 and 36 is
Answer: 12.

Question. The HCF of 45 and 105 is 15. Write their LCM.
Answer: \( \text{LCM} = \frac{45 \times 105}{15} = 315 \).

Question. Find pairs of natural numbers whose least common multiple is 78 and the greatest divisor is 13.
Answer: (13, 78) and (26, 39).

Question. Prove that \( 3 + \sqrt{2} \) is irrational number, given that \( \sqrt{2} \) is an irrational number.
Answer: Let \( 3 + \sqrt{2} = \frac{a}{b} \implies \sqrt{2} = \frac{a}{b} - 3 = \frac{a - 3b}{b} \). Since a and b are integers, the RHS is rational, which contradicts that \( \sqrt{2} \) is irrational. Hence \( 3 + \sqrt{2} \) is irrational.

Question. Find the HCF and LCM of 288, 360 and 384 by prime factorisation method.
Answer:
\( 288 = 2^5 \times 3^2 \)
\( 360 = 2^3 \times 3^2 \times 5 \)
\( 384 = 2^7 \times 3 \)
HCF = \( 2^3 \times 3 = 24 \).
LCM = \( 2^7 \times 3^2 \times 5 = 128 \times 9 \times 5 = 5760 \).

Question. If x is an even number, then what is the LCM of 4x, \( 2x^2 \) and \( x^3 \)?
Answer: LCM is \( 2x^3 \).

Question. The HCF of 24 and 36 is d. Find two numbers a and b, such that \( d = 24a + 36b \). 
Answer: \( d = 12 \). One pair is \( a = -1, b = 1 \) (\( 12 = -24 + 36 \)).

Question. The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm respectively. Determine the length of the longest rod which can measure the three dimensions of the room exactly.
Answer: Dimensions: 825, 675, 450 cm. HCF(825, 675, 450) = 75 cm.

Question. On a morning walk three persons step off together and their steps measure 40 cm, 42 cm, 45 cm, what is the minimum distance each should walk so that each can cover the same distance in complete steps?
Answer: LCM(40, 42, 45) = 2520 cm = 25.2 m.

Question. P is a prime and Q is a positive integer such that P + Q = 1696. If P and Q are co-prime and their LCM is 21879, then find P and Q.
Answer: Since P and Q are co-prime, \( \text{LCM}(P, Q) = PQ = 21879 \). Also \( P + Q = 1696 \).
Solving the quadratic \( x^2 - 1696x + 21879 = 0 \):
Factors are 13 and 1683. Since P is prime, \( P = 13 \).
So, \( P = 13, Q = 1683 \).

Question. Let d be the HCF of 24 and 36. Find two numbers a and b, such that \( d = 24a + 36b \). 
Answer: \( d = 12 \). For \( 12 = 24a + 36b \), possible values are \( a = -1, b = 1 \).

Question. Find the least number which when divided by 16, leaves a remainder 6, when divided by 19 leaves a remainder 9 and when divided by 21 leaves a remainder 11.
Answer: Difference \( 16 - 6 = 10 \), \( 19 - 9 = 10 \), \( 21 - 11 = 10 \).
Required number = LCM(16, 19, 21) - 10 = 6384 - 10 = 6374.

Question. Two natural numbers whose least common multiple is 78 and the greatest divisor is 13.
Answer: The pairs are (13, 78) and (26, 39).

Question. Show that \( 21^n \) can not end with the digits 0, 2, 4, 6 and 8 for any natural number n. 
Answer: \( 21^n = (3 \times 7)^n \). The last digit of power of numbers ending in 1 is always 1.