📘 NCERT · Class 10 · Maths

Chapter 1 — Real Numbers MCQ

✅ 45 Questions 💡 With Solutions 📖 NCERT Based 🆓 Free

Here are 45 important MCQs from Class 10 Maths Chapter 1 — Real Numbers, based on the NCERT textbook syllabus. Each question is accompanied by its correct answer and step-by-step solution. Absolutely perfect for board exam preparation!

📝 Chapter Summary — Real Numbers

Real Numbers is the first chapter of Class 10. In this chapter, we study Euclid's Division Lemma, Fundamental Theorem of Arithmetic, and Rational vs Irrational Numbers. This chapter forms the foundation of the entire Number System.

Euclid's Division Lemma: For every positive integer a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.
Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed as a product of prime numbers (irrespective of order).
HCF and LCM: For any two positive integers a and b, HCF(a,b) × LCM(a,b) = a × b.
Irrational Numbers: √2, √3, √5 and the square root of any prime is always irrational — this is proved using the method of contradiction.
Decimal Expansion of Rational Numbers: If the denominator has only 2 and 5 as prime factors, the decimal will be terminating; otherwise it will be non-terminating repeating.
Decimal Expansion of Irrational Numbers: Always non-terminating and non-repeating, such as π, √2, √3.
Co-prime Numbers: Two numbers are co-prime if their HCF = 1, for example 4 and 9.
Rational + Irrational = Irrational: The sum or product of any non-zero rational number with an irrational number is always irrational.

📊 Quick Poll — Apni Raay Do!

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Euclid's Division Lemma

Fundamental Theorem of Arithmetic

Irrational Numbers ka Proof

Terminating vs Non-terminating Decimals

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❓ 45 MCQs — Real Numbers (NCERT Class 10)

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of:

A Even numbers

B Prime numbers

C Composite numbers

D Natural numbers

✓ Correct Answer: B

Solution: Every composite number can be uniquely factorized into a product of primes, regardless of the order. This is the core of the Fundamental Theorem of Arithmetic.

Which of the following is an irrational number?

A 3.14

B 22/7

C √2

D 0.333...

✓ Correct Answer: C

Solution: √2 is irrational because 2 is a non-perfect square. The square root of any non-perfect square is always irrational. 0.333... = 1/3 is rational.

If p is a prime number and p divides a², where a is a positive integer, then:

A p must divide a

B p² must divide a

C a must be prime

D p cannot divide a

✓ Correct Answer: A

Solution: According to a key theorem in NCERT, if a prime p divides a², then p must also divide a. This theorem is used to prove that certain numbers are irrational.

The HCF of 96 and 404 is:

A 2

B 4

C 96

D 101

✓ Correct Answer: B

Solution: 96 = 2⁵ × 3, and 404 = 2² × 101. The only common prime factor is 2 with minimum power 2² = 4. Therefore HCF = 4.

The exponent of 2 in the prime factorization of 144 is:

A 2

B 3

C 4

D 5

✓ Correct Answer: C

Solution: 144 = 12² = (2² × 3)² = 2⁴ × 3². Isliye 2 ka exponent 4 hai.

LCM(a, b) × HCF(a, b) is equal to:

A a + b

B a − b

C a × b

D a ÷ b

✓ Correct Answer: C

Solution: For any two positive integers a and b: LCM(a,b) × HCF(a,b) = a × b. This is a very important formula that frequently appears in board exams.

The HCF of two co-prime numbers is always:

A 0

B 1

C Their product

D Their sum

✓ Correct Answer: B

Solution: There is only 1 common factor between co-prime numbers. Therefore their HCF = 1. Example: 4 and 9 are co-prime, HCF(4,9) = 1.

Which of these is the prime factorization of 156?

A 2 × 3 × 13

B 2² × 3 × 13

C 2 × 3² × 13

D 2² × 3² × 13

✓ Correct Answer: B

Solution: 156 = 2 × 78 = 2 × 2 × 39 = 4 × 39 = 4 × 3 × 13 = 2² × 3 × 13. Confirm using a factor tree.

If HCF(306, 657) = 9, then LCM(306, 657) is:

A 22338

B 22383

C 23238

D 22339

✓ Correct Answer: A

Solution: LCM = (a × b) ÷ HCF = (306 × 657) ÷ 9 = 201042 ÷ 9 = 22338.

Q10

The decimal expansion of √5 is:

A Terminating

B Non-terminating repeating

C Non-terminating non-repeating

D Terminating repeating

✓ Correct Answer: C

Solution: √5 is an irrational number. The decimal expansion of irrational numbers is always non-terminating and non-repeating. √5 ≈ 2.2360679...

Q11

The largest number that divides 70 and 125, leaving remainders 5 and 8 respectively, is:

A 13

B 65

C 875

D 1750

✓ Correct Answer: A

Solution: That number will be HCF(70−5, 125−8) = HCF(65, 117). 65 = 5×13, 117 = 9×13. HCF = 13.

Q12

The product of a non-zero rational and an irrational number is:

A Always rational

B Always irrational

C Rational or irrational

D One

✓ Correct Answer: B

Solution: If rational r ≠ 0 and x is irrational, then r×x will always be irrational. Example: 3 × √2 = 3√2 (irrational).

Q13

For any positive integer n, 6ⁿ can end with the digit 0 if its prime factorization contains:

A Only 2

B Only 5

C Both 2 and 5

D Neither 2 nor 5

✓ Correct Answer: C

Solution: Any number ends in 0 only when both 2 and 5 are present in its prime factors. 6ⁿ = 2ⁿ × 3ⁿ — it has no factor of 5, so 6ⁿ will never end in 0.

Q14

The sum of a rational and an irrational number is:

A Rational

B Irrational

C Zero

D Integer

✓ Correct Answer: B

Solution: The sum of a rational and an irrational number is always irrational. Example: 2 + √3 is irrational. This is demonstrated in NCERT using proof by contradiction.

Q15

The HCF of 26 and 91 is:

A 7

B 13

C 26

D 1

✓ Correct Answer: B

Solution: 26 = 2 × 13 and 91 = 7 × 13. Common factor = 13. Therefore HCF(26, 91) = 13.

Q16

If a = x³y² and b = xy³, where x, y are prime numbers, then HCF(a, b) is:

A xy

B xy²

C x³y³

D x²y²

✓ Correct Answer: B

Solution: For HCF, we take the lowest power of common factors. For x: min(3,1) = 1, for y: min(2,3) = 2. HCF = x¹y² = xy².

Q17

If a = x³y² and b = xy³, where x, y are prime numbers, then LCM(a, b) is:

A x³y³

B xy

C x²y²

D x³y²

✓ Correct Answer: A

Solution: For LCM, we take the highest power. For x: max(3,1) = 3, for y: max(2,3) = 3. LCM = x³y³.

Q18

Which of the following is NOT a prime factor of 3825?

A 3

B 5

C 17

D 11

✓ Correct Answer: D

Solution: 3825 = 3 × 1275 = 3 × 3 × 425 = 9 × 425 = 9 × 5 × 85 = 9 × 5 × 5 × 17 = 3² × 5² × 17. 11 does not appear in any step.

Q19

The number (√3 + √2)(√3 − √2) is:

A Rational

B Irrational

C Complex

D Not defined

✓ Correct Answer: A

Solution: (√3 + √2)(√3 − √2) = (√3)² − (√2)² = 3 − 2 = 1. Yeh 1 hai jo rational number hai. (a+b)(a−b) = a² − b² formula use karo.

Q20

The product of two consecutive positive integers is always divisible by:

A 2

B 3

C 4

D 5

✓ Correct Answer: A

Solution: Among two consecutive integers, one is always even. An even number is divisible by 2. Therefore their product will always be divisible by 2.

Q21

If the LCM of 12 and 42 is 10m + 4, then the value of m is:

A 7

B 8

C 9

D 10

✓ Correct Answer: B

Solution: LCM(12, 42): 12 = 2²×3, 42 = 2×3×7. LCM = 2²×3×7 = 84. Now 10m + 4 = 84 ⟹ 10m = 80 ⟹ m = 8.

Q22

π is:

A An integer

B A rational number

C An irrational number

D A terminating decimal

✓ Correct Answer: C

Solution: π is an irrational number. Its decimal expansion is non-terminating and non-repeating: π ≈ 3.14159265... Note: 22/7 is only an approximation, not equal to π.

Q23

The HCF of 8, 9, 25 is:

A 1

B 5

C 10

D 25

✓ Correct Answer: A

Solution: 8 = 2³, 9 = 3², 25 = 5². There is no common prime factor among the three. Therefore HCF = 1 — these three numbers are mutually co-prime.

Q24

The number of prime factors of 105 is:

A 2

B 3

C 4

D 5

✓ Correct Answer: B

Solution: 105 = 3 × 35 = 3 × 5 × 7. There are three distinct prime factors: 3, 5, and 7.

Q25

The product of three consecutive positive integers is divisible by:

A 4

B 6

C 8

D 10

✓ Correct Answer: B

Solution: Among three consecutive integers, one must be divisible by 2 and one by 3. Therefore the product is always divisible by 2×3 = 6. Example: 4×5×6 = 120 = 6×20.

Q26

The HCF of the smallest composite number and the smallest prime number is:

A 1

B 2

C 3

D 4

✓ Correct Answer: B

Solution: The smallest composite number = 4, the smallest prime = 2. HCF(4, 2) = 2.

Q27

2 + √3 is:

A A rational number

B An irrational number

C An integer

D A whole number

✓ Correct Answer: B

Solution: 2 is rational and √3 is irrational. Rational + Irrational = Irrational. Therefore 2 + √3 is irrational.

Q28

A rational number can be expressed as p/q where p and q are integers and q is:

A Equal to zero

B Not equal to zero

C Greater than one

D Less than zero

✓ Correct Answer: B

Solution: The definition of a rational number is: p/q where p and q are integers and q ≠ 0. If q = 0, the fraction becomes undefined.

Q29

The exponent of 5 in the prime factorization of 3750 is:

A 3

B 4

C 5

D 6

✓ Correct Answer: B

Solution: 3750 = 2 × 1875 = 2 × 3 × 625 = 2 × 3 × 5⁴. Isliye 5 ka exponent 4 hai.

Q30

If the HCF of 65 and 117 is expressible in the form 65m − 117, then the value of m is:

A 1

B 2

C 3

D 4

✓ Correct Answer: B

Solution: HCF(65, 117): 65 = 5×13, 117 = 9×13. HCF = 13. Now 65m − 117 = 13 ⟹ 65m = 130 ⟹ m = 2.

Q31

The product of two irrational numbers is:

A Always rational

B Always irrational

C Sometimes rational, sometimes irrational

D Zero

✓ Correct Answer: C

Solution: √2 × √2 = 2 (rational); √2 × √3 = √6 (irrational). Both cases are possible, so the answer is "sometimes rational, sometimes irrational".

Q32

Is 7 × 11 × 13 + 13 a composite number?

A Yes

B No

C Cannot be determined

D Only if divided by 13

✓ Correct Answer: A

Solution: 7×11×13 + 13 = 13(7×11 + 1) = 13 × (77 + 1) = 13 × 78. It has factors: 1, 2, 3, 6, 13, 26... This is clearly composite.

Q33

The HCF of 135 and 225 is:

A 15

B 30

C 45

D 75

✓ Correct Answer: C

Solution: 135 = 3³ × 5 = 27 × 5, and 225 = 3² × 5² = 9 × 25. HCF = 3² × 5¹ = 9 × 5 = 45.

Q34

What is the LCM of 12, 15 and 21?

A 420

B 120

C 240

D 360

✓ Correct Answer: A

Solution: 12 = 2² × 3, 15 = 3 × 5, 21 = 3 × 7. LCM = 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420.

Q35

Which of the following is co-prime to 18?

A 24

B 35

C 21

D 27

✓ Correct Answer: B

Solution: 18 = 2 × 3². HCF(18, 35): 35 = 5 × 7. 18 and 35 have no common prime factor. HCF = 1. Others: 24 shares 2, 21 shares 3, 27 shares 3.

Q36

If n is a natural number, then 9ⁿ − 5ⁿ is always divisible by:

A 2

B 4

C 9

D 5

✓ Correct Answer: B

Solution: xⁿ − yⁿ is always divisible by (x − y). Here x − y = 9 − 5 = 4. Therefore 9ⁿ − 5ⁿ will always be divisible by 4.

Q37

HCF of 72 and 120 is:

A 12

B 24

C 36

D 48

✓ Correct Answer: B

Solution: 72 = 2³ × 3² and 120 = 2³ × 3 × 5. HCF = 2³ × 3¹ = 8 × 3 = 24.

Q38

√p where p is prime is always:

A Rational

B Irrational

C Integer

D Whole number

✓ Correct Answer: B

Solution: The square root of any prime number is irrational. This is proved in NCERT using the contradiction method. √2, √3, √5, √7 — all are irrational.

Q39

If p/q is a rational number, then the condition for it to have a terminating decimal expansion is q =

A 2ⁿ

B 5ᵐ

C 2ⁿ5ᵐ

D 3ⁿ

✓ Correct Answer: C

Solution: NCERT theorem: The decimal of p/q will be terminating if the prime factors of q are only 2 and 5, i.e., q is of the form 2ⁿ × 5ᵐ.

Q40

Which of the following is a rational number?

A √2

B √3

C √4

D √5

✓ Correct Answer: C

Solution: √4 = 2, which is an integer and a rational number. √2, √3, √5 — all are irrational because 2, 3, 5 are non-perfect squares.

Q41

The number 3 × 5 × 7 + 7 is:

A Prime

B Composite

C Irrational

D Odd

✓ Correct Answer: B

Solution: 3 × 5 × 7 + 7 = 7(3 × 5 + 1) = 7 × 16 = 112. Factors of 112 are 1, 2, 4, 7, 8, 14, 16, 28, 56, 112. Clearly composite.

Q42

Which is irrational?

A 0.14

B 0.1416

C 0.141414...

D 0.4010010001...

✓ Correct Answer: D

Solution: 0.14 aur 0.1416 terminating hain (rational). 0.141414... = 0.14̄ repeating hai (rational). 0.4010010001... mein pattern change hota rehta hai — yeh non-repeating irrational hai.

Q43

If HCF(a, b) = 12 and a × b = 1800, then LCM(a, b) is:

A 150

B 180

C 200

D 1500

✓ Correct Answer: A

Solution: Formula: LCM × HCF = a × b. LCM = (a × b) ÷ HCF = 1800 ÷ 12 = 150.

Q44

The decimal form of 7/8 is:

A 0.875

B 0.75

C 0.825

D 0.625

✓ Correct Answer: A

Solution: 8 = 2³ (only a power of 2) — so it will be a terminating decimal. 7 ÷ 8 = 0.875. Verify: 0.875 × 8 = 7 ✓

Q45

The HCF of 441, 567 and 693 is:

A 63

B 441

C 189

D 21

✓ Correct Answer: A

Solution: 441 = 63 × 7, 567 = 63 × 9, 693 = 63 × 11. All three numbers are multiples of 63. HCF = 63.

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