Class 10 Maths Real Numbers MCQ

Class 10 Mathematics — Real Numbers

Quality Question Bank

A. MCQs

1. The prime factorisation of a number is unique except for:

A. the powers of the primes
B. the order of the prime factors
C. the number of factors
D. the largest prime factor

2. IfN=24×32×5,

then the HCF of N and 22×34×7 is:

A. 22×32
B. 24×32
C. 22×34
D. 24×34

3. Ifa=23×32,b=22×34×5,

then LCM(a,b) is:

A. 22×32×5
B. 23×34×5
C. 23×32
D. 25×36×5

4. Which statement is necessarily true?

A. If p∣a2, then p∣a2 only when a is prime.
B. If p is prime and p∣a2, then p∣a.
C. Every divisor of a2 divides a.
D. If p∣a, then a∣p.

5. Which number cannot be the last digit of 4n, where n is a natural number?

A. 2
B. 4
C. 6
D. 8

6. If a and b are positive integers andHCF(a,b)=6,ab=540,

then their LCM is:

A. 60
B. 90
C. 3240
D. 546

7. Which statement gives the correct criterion for HCF using prime factorisation?

A. Use the greatest powers of all primes.
B. Use the smallest powers of common primes.
C. Use only the largest prime factor.
D. Use every prime occurring in either number with its smallest power.

8. If p is prime and p∣a2, the result used in proving irrationality of p is:

A. p∣a
B. a∣p
C. p=a
D. a2=p

9. Supposep=ba

where a,b are coprime integers and b=0. After squaring, what relation is obtained?

A. pb=a2
B. pb2=a2
C. p2b=a
D. pa2=b2

10. In a contradiction proof for the irrationality of 2, assuming2=ba

ultimately forces:

A. only a to be even
B. only b to be even
C. both a and b to be even
D. neither a nor b to be even

11. If r is a non-zero rational number and x is irrational, which is necessarily irrational?

A. rx
B. r+x
C. r−x
D. All of these

12. Which assumption is essential in the standard proof of irrationality of 2?

A. Numerator and denominator are both prime
B. Numerator and denominator are coprime
C. Numerator is greater than denominator
D. Denominator is prime

B. Assertion–Reason

For each question, choose:

A. Both A and R are true, and R correctly explains A
B. Both A and R are true, but R does not correctly explain A
C. A is true, but R is false
D. A is false, but R is true

13.
Assertion (A): The prime factorisation of a composite number is unique apart from the order of its factors.
Reason (R): Fundamental Theorem of Arithmetic guarantees uniqueness of prime factorisation.

14.
Assertion (A): If p is prime and p∣a2, then p∣a.
Reason (R): Every prime factor appearing in a2 already appears in the prime factorisation of a.

15.
Assertion (A): If a and b are positive integers, thenHCF(a,b)LCM(a,b)=ab.

Reason (R): In the prime factorisations, the smaller and larger powers of every prime together reproduce the powers occurring in ab.

16.
Assertion (A): 4n cannot end in zero for any natural number n.
Reason (R): The prime factorisation of 4n contains only the prime 2.

17.
Assertion (A): 2+5 is irrational.
Reason (R): If 2+5 were rational, subtracting the rational number 5 would make 2 rational.

C. Fill in the Blanks

18. Every composite number can be expressed as a product of ______ numbers.

19. The prime factorisation of a natural number is unique except for the ______ of its prime factors.

20. For two positive integers a,b,HCF(a,b)×LCM(a,b)=.

21. To obtain the HCF from prime factorisations, we select the ______ powers of the common prime factors.

22. To obtain the LCM, we select the ______ powers of all prime factors involved.

23. If p is prime and p∣a2, then p∣.

24. A number that cannot be expressed as p/q, where p,q are integers and q=0, is called a(n) ______ number.

25. The standard proof that 2 is irrational uses proof by ______.

26. In the contradiction proof, the numerator and denominator are taken to be ______.

D. True or False — With Correction

State whether each statement is True or False. If false, rewrite it correctly.

27. The LCM is obtained using the smallest powers of the prime factors.

28. The HCF of three positive integers multiplied by their LCM is always equal to their product.

29. If a prime divides the square of an integer, it must divide the integer itself.

30. The uniqueness in prime factorisation means that the order of prime factors must always remain fixed.

31. If x is irrational, then x+2 must be rational.

32. In proving 2 irrational, assuming it is rational eventually leads to a contradiction with the coprimality of the numerator and denominator.

E. Match the Following

33. Match Column I with Column II.

Column I	Column II
(a) HCF	(i) Greatest powers
(b) LCM	(ii) Proof by contradiction
(c) Irrationality proof	(iii) Smallest powers of common primes
(d) Fundamental Theorem of Arithmetic	(iv) Unique prime factorisation

F. Very Short Answer — Conceptual

34. Why does the prime factorisation of 4n prevent 4n from being divisible by 5?

35. Why is the condition that the numerator and denominator be coprime important when proving 2 irrational?

36. What contradiction is obtained at the end of the proof that 2 is irrational?

37. Why can the HCF–LCM product relation for two numbers not simply be extended to three numbers?

38. What role does uniqueness of prime factorisation play in the theorem p∣a2⇒p∣a?

G. Short Answer — Higher-Order

39. LetA=23×32×5,B=22×34×7.

Without multiplying A or B, determine their HCF and LCM.

40. Two positive integers have HCF 12 and product 1728. Determine their LCM.

41. Explain why a number whose prime factorisation contains no factor 5 cannot have 0 as its units digit.

42. Suppose a proof begins with the assumptionp=ba,

where p is prime and a,b are coprime. Explain the chain of reasoning that should lead to a contradiction.

43. A student claims:
“Since 3 is irrational, 3+5 must also be irrational.”
Identify the error in the reasoning and state the correct principle involving an irrational number and a rational number.

H. Case-Based Questions

Case 1: Prime Factorisation

A student writesN=24×32×53

and another number asM=22×34×7.

44. Find HCF(N,M).

45. Find LCM(N,M).

46. Which prime factors occur in the LCM but not in the HCF?

47. Without calculating N and M, determine whether their product is divisible by 26.

Case 2: Irrationality Proof

A student wants to prove that 5 is irrational. They assume5=ba

where a,b are coprime and b=0.

48. What equation results after squaring?

49. Which theorem can be applied after establishing that 5 divides a2?

50. What does this theorem imply about a?

51. What further conclusion eventually contradicts the assumption that a,b are coprime?

I. HOTS / Reasoning Questions

52. A positive integer has prime factorisation2a3b5c.

For its square to be divisible by 5, what condition must c satisfy? Explain using prime factorisation.

53. IfHCF(a,b)=1,

what does this tell you about the prime factors common to a and b? How is this useful in contradiction proofs?

54. A number N has exactly the prime factors 2,3, and 7. Another number M has exactly the prime factors 2,5, and 7. Which prime factors must occur in their LCM, and which can occur in their HCF?

55. Explain why the following style of argument proves irrationality:

Assume an irrational-looking number is rational, derive that a known irrational number would then be rational, and obtain a contradiction.

Illustrate the reasoning using either 3−5 or 32.

Answer Key

MCQs

Assertion–Reason

Fill in the blanks

prime
order
ab
smallest
greatest
a
irrational
contradiction
coprime

True/False

False — LCM uses the greatest powers.
False — the two-number HCF–LCM product relation does not generally extend to three numbers.
True
False — the order may change; the prime factors themselves and their multiplicities are uniquely determined.
False — irrational + rational is irrational.
True

Match

(a)–(iii), (b)–(i), (c)–(ii), (d)–(iv)