Real Numbers is one of the foundation chapters of Class 10 Mathematics because it helps students understand important number system concepts used in advanced maths topics. Practicing Real Numbers Class 10 MCQs helps students revise concepts like Euclid’s Division Lemma, HCF and LCM, prime factorization, rational numbers, irrational numbers, and decimal expansion in a better way.
These questions are prepared to improve conceptual understanding, calculation accuracy, and problem-solving skills. Students can also practice more MCQs from different chapters, explore complete Class 10 MCQs for board exam preparation, and solve chapter-wise Class 10 Maths MCQs to strengthen their overall preparation.
Each question includes the correct answer with a detailed explanation so that students can understand the reasoning behind every solution instead of just memorizing answers. These Real Numbers objective questions are helpful for quick revision, school exams, CBSE board exams, and competency-based question practice.
Important Topics Covered in Real Numbers Class 10 MCQs
Before solving the questions, revise the major concepts from Chapter 1 because most MCQs are based on formulas, applications, and logical understanding.
- Euclid’s Division Lemma
- Euclid’s Division Algorithm
- Finding HCF using division method
- Fundamental Theorem of Arithmetic
- Prime factorization of numbers
- Relationship between HCF and LCM
- Rational and irrational numbers
- Decimal expansion of rational numbers
- Terminating decimals
- Non-terminating recurring decimals
- Co-prime numbers
Understanding these topics will help you solve both direct and competency-based Real Numbers MCQs more confidently.
Real Numbers Class 10 MCQs with Answers
Practice these important Real Numbers Class 10 MCQs designed according to the latest exam pattern. These questions cover important concepts from CBSE Class 10 Maths Chapter 1 and will help improve accuracy, revision speed, and conceptual understanding.
Q. If two positive integers a and b are written as a = x³y² and b = xy³, where x and y are distinct prime numbers, then what is the HCF(a, b)?
(A) xy
(B) x³y³
(C) x²y²
(D) xy²
Answer: D
Explanation:
To find the Highest Common Factor (HCF), we find the product of the lowest power of each common prime factor involved in the numbers. For prime factor x, the lowest power between x³ and x¹ is x¹. For prime factor y, the lowest power between y² and y³ is y². Thus, HCF(a, b) = x¹ × y² = xy².
Q. According to the Fundamental Theorem of Arithmetic, every composite number can be uniquely expressed as a product of primes, disregarding what property?
(A) The values of the prime factors
(B) The order in which the prime factors occur
(C) The total number of factors
(D) The exponents of the prime factors
Answer: B
Explanation:
The Fundamental Theorem of Arithmetic states that every composite number can be factored into a unique product of prime numbers, up to the order in which these factors are arranged. Changing the order (e.g., 2 × 3 vs 3 × 2) does not change the core unique factorization.
Q. If the LCM of two numbers is 360, which of the following can never be their HCF?
(A) 16
(B) 24
(C) 60
(D) 90
Answer: A
Explanation:
The HCF of any set of numbers must always be a perfect divisor of their LCM. Let us test the options: 360 ÷ 24 = 15; 360 ÷ 60 = 6; 360 ÷ 90 = 4. However, 360 ÷ 16 = 22.5 (not an integer). Therefore, 16 can never be the HCF.
Q. Let n be an arbitrary natural number. Then the expression 6ⁿ – 5ⁿ always terminates with which digit?
(A) 0
(B) 1
(C) 5
(D) 6
Answer: B
Explanation:
For any natural number n, 6ⁿ always ends in the digit 6 (since 6×6=36, 36×6=216, etc.), and 5ⁿ always ends in the digit 5 (5, 25, 125, etc.). Subtracting a number ending in 5 from a number ending in 6 will always yield a units digit of (6 – 5) = 1.
Q. If p is a prime number and it divides a² (where a is a positive integer), which statement is logically guaranteed by fundamental number theory?
(A) p must divide a
(B) a must be a prime number
(C) p² must divide a
(D) p cannot divide a
Answer: A
Explanation:
A standard theorem in real number theory dictates that if a prime number p divides a square integer a², then p must also divide the base integer a. This forms the mathematical backbone for proving the irrationality of numbers like √2 or √3.
Q. What is the smallest positive integer that is perfectly divisible by all integers spanning from 1 to 10 inclusive?
(A) 100
(B) 1260
(C) 2520
(D) 5040
Answer: C
Explanation:
To find this integer, we calculate the LCM of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Prime factorizations are: 2=2, 3=3, 4=2², 5=5, 6=2×3, 7=7, 8=2³, 9=3², 10=2×5. Taking highest powers: LCM = 2³ × 3² × 5 × 7 = 8 × 9 × 5 × 7 = 2520.
Q. If the HCF of 65 and 117 is written in the algebraic form 65m – 117, what is the value of m?
(A) 1
(B) 2
(C) 3
(D) 4
Answer: B
Explanation:
First find HCF(65, 117). 65 = 5 × 13, and 117 = 9 × 13 = 3² × 13. The common factor is 13, so HCF = 13. Set up the equation: 65m – 117 = 13 => 65m = 130 => m = 130 / 65 = 2.
Q. Which of the following numbers is an irrational number?
(A) 3.14159
(B) 22/7
(C) 0.120120012000...
(D) 4.316316316...
Answer: C
Explanation:
Irrational numbers are characterized by non-terminating, non-repeating decimal representations. Option (A) terminates, (B) is a fraction p/q, (D) is non-terminating but repeating (periodic). Option (C) displays a changing non-repeating pattern, which confirms its irrationality.
Q. The product of a non-zero rational number and an irrational number is always guaranteed to be:
(A) A rational number
(B) An irrational number
(C) An integer
(D) Zero
Answer: B
Explanation:
Let r be a non-zero rational number and x be an irrational number. If their product rx were rational (say, k), then x = k/r would be rational, creating a contradiction. Hence, the product must be irrational.
Q. If p₁ and p₂ are two distinct odd prime numbers such that p₁ > p₂, then the expression (p₁² – p₂²) is always:
(A) An odd prime number
(B) An odd integer but not prime
(C) An even integer divisible by 8
(D) An even integer but not divisible by 4
Answer: C
Explanation:
Any odd prime square leaves a remainder of 1 when divided by 8 (since all odd squares are of the form 8k+1). Thus, p₁² = 8k+1 and p₂² = 8j+1. Subtracting them yields: (8k+1) – (8j+1) = 8(k – j), proving it is always fully divisible by 8.
Q. The HCF of two numbers is 9 and their LCM is 459. If one of the numbers is 27, calculate the value of the other number.
(A) 153
(B) 81
(C) 243
(D) 51
Answer: A
Explanation:
Use the fundamental identity: Product of two numbers = HCF × LCM. Let the missing number be x. 27 × x = 9 × 459 => x = (9 × 459) / 27 => x = 459 / 3 = 153.
Q. What is the total number of prime factors present in the prime factorization of 5005?
(A) 2
(B) 3
(C) 4
(D) 5
Answer: C
Explanation:
Perform systematic prime factorization on 5005: It ends in 5, so 5005 ÷ 5 = 1001. 1001 is divisible by 7 (1001 ÷ 7 = 143). 143 is divisible by 11 (143 ÷ 11 = 13). 13 is prime. So 5005 = 5 × 7 × 11 × 13. There are exactly 4 unique prime factors.
Q. If x and y are two co-prime integers, then what must be the value of their LCM?
(A) 1
(B) x + y
(C) xy
(D) x / y
Answer: C
Explanation:
Co-prime numbers are numbers whose only common positive factor is 1 (HCF = 1). Because HCF × LCM = Product of Numbers, we get 1 × LCM = x × y => LCM = xy.
Q. For any two positive integers a and b, what is the exact logical relationship between HCF(a, b) and LCM(a, b)?
(A) HCF strictly exceeds LCM
(B) LCM is a factor of HCF
(C) HCF is a factor of LCM
(D) They never share any factors
Answer: C
Explanation:
By definitions of factors and multiples, the Highest Common Factor encapsulates the shared modular units, whereas the Least Common Multiple pools all constituent prime parts. Therefore, the HCF will always perfectly divide the LCM.
Q. If an item occurs in repeating intervals of 12 minutes, 15 minutes, and 18 minutes, and they all trigger together at 9:00 AM, at what time will they next trigger simultaneously?
(A) 10:00 AM
(B) 11:00 AM
(C) 12:00 PM
(D) 1:00 PM
Answer: C
Explanation:
To find the next shared interval, calculate the LCM of 12, 15, and 18. 12 = 2²×3, 15 = 3×5, 18 = 2×3². LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180 minutes. 180 minutes equals exactly 3 hours. 9:00 AM + 3 hours = 12:00 PM.
Q. What is the exponent of the prime factor 2 in the complete prime factorization of 144?
(A) 2
(B) 3
(C) 4
(D) 5
Answer: C
Explanation:
Dividing 144 sequentially by 2 gives: 144→72→36→18→9. Since 9 is odd, we stop. We divided by 2 exactly 4 times. Thus, 144 = 2⁴ × 3², meaning the exponent of 2 is 4.
Q. If n is a positive integer, then n² – n is always divisible by which number?
(A) 2
(B) 3
(C) 4
(D) 5
Answer: A
Explanation:
We can factor n² – n as n(n – 1). This represents the product of two consecutive integers. Out of any two consecutive integers, one must be even. Therefore, the product is always divisible by 2.
Q. What type of number is represented by the expression (3 + 2√5) – (2 + 2√5)?
(A) Irrational number
(B) Rational number
(C) Complex imaginary number
(D) Fractional non-integer
Answer: B
Explanation:
Expanding the expression: 3 + 2√5 – 2 – 2√5. The irrational parts (+2√5 and –2√5) cancel out completely, leaving 3 – 2 = 1. Since 1 is an integer, it is a rational number.
Q. If HCF(a, b) = 1, what unique identifier name is assigned to this pair of integers a and b?
(A) Composite pairs
(B) Perfect matches
(C) Co-prime numbers
(D) Twin primes
Answer: C
Explanation:
By definition, integers a and b are called co-prime (or relatively prime) if their highest common factor is 1. They do not need to be individual prime numbers themselves (e.g., 8 and 9 are co-prime).
Q. The sum of a rational number and an irrational number is always:
(A) Rational
(B) Irrational
(C) An Integer
(D) Determinable only case-by-case
Answer: B
Explanation:
Adding a repeating/terminating decimal to a non-repeating non-terminating decimal always yields a non-repeating non-terminating decimal. Hence, the sum is strictly irrational.
Q. Which of the following values matches the expression HCF(a, b) × LCM(a, b)?
(A) a + b
(B) a – b
(C) a × b
(D) a ÷ b
Answer: C
Explanation:
For any two positive integers a and b, the product of their Highest Common Factor and Least Common Multiple is exactly equal to the product of the two numbers themselves.
Q. If p is a prime number, what is the LCM of p, p², and p³?
(A) p
(B) p²
(C) p³
(D) p⁶
Answer: C
Explanation:
To find the Least Common Multiple from prime bases, we select the factor with the highest power among the expressions. The highest power among p¹, p², and p³ is p³.
Q. What is the HCF of the smallest prime number and the smallest composite number?
(A) 1
(B) 2
(C) 4
(D) 8
Answer: B
Explanation:
The smallest prime number is 2. The smallest composite number is 4. Finding the common factors: 2 = 2¹, and 4 = 2². The highest common factor is 2¹ = 2.
Q. If x is a rational number and y is an irrational number, what can you conclude about the value of x – y?
(A) It is always rational
(B) It is always irrational
(C) It alternates between both forms
(D) It equals zero
Answer: B
Explanation:
Similar to addition, subtracting an irrational number from a rational number preserves the infinite, non-repeating properties of the digits. Therefore, the difference is strictly irrational.
Q. What is the maximum number of times any two distinct straight lines can cross or intersect on a flat Cartesian real coordinate plane?
(A) 0
(B) 1
(C) 2
(D) Infinite
Answer: B
Explanation:
Two distinct lines are either parallel (intersect 0 times) or non-parallel. Non-parallel lines cross exactly once. Therefore, the maximum number of intersection points is 1.
Q. If a positive number can be written as 4ᵏ, with what digits can it never terminate for any natural power k?
(A) 4
(B) 6
(C) 0
(D) Both A and B
Answer: C
Explanation:
Evaluating powers of 4: 4¹=4, 4²=16, 4³=64, 4⁴=256. The values cycle strictly between ending in 4 and 6. Consequently, a power of 4 can never end in 0.
Q. Which of the following is a correct prime factorization of the integer 120?
(A) 2 × 3 × 4 × 5
(B) 2² × 3² × 5
(C) 2³ × 3 × 5
(D) 8 × 15
Answer: C
Explanation:
Prime factorization requires all constituent bases to be prime numbers. 120 = 2 × 60 = 2 × 2 × 30 = 2 × 2 × 2 × 15 = 2³ × 3 × 5. Option A and D contain composite numbers, making them invalid prime factorizations.
Q. What is the HCF of any two consecutive odd natural numbers?
(A) 1
(B) 2
(C) 3
(D) Variable
Answer: A
Explanation:
Consecutive odd numbers (like 11 and 13, or 25 and 27) share no common factor other than 1. Therefore, consecutive odd integers are always co-prime, meaning their HCF is 1.
Q. What is the mathematical definition of a rational number?
(A) A number that cannot be written as a fraction
(B) A number written as p/q, where p and q are integers and q is non-zero
(C) A number whose decimal expansion never ends
(D) The square root of a prime number
Answer: B
Explanation:
A rational number is formally defined as any number that can be written in the fraction format p/q, where the numerator p and denominator q are integers, and q is not equal to 0.
Q. If the mathematical operation HCF(k, 8) = 4 and LCM(k, 8) = 24, find the value of the unknown integer k.
(A) 6
(B) 12
(C) 16
(D) 20
Answer: B
Explanation:
Using the identity HCF × LCM = Product of Numbers: 4 × 24 = k × 8 => 96 = 8k => k = 96 ÷ 8 = 12.
Important Formulas and Rules for Real Numbers MCQs
A quick revision of these formulas and concepts can help avoid common mistakes while solving objective questions.
Euclid’s Division Lemma
For any two positive integers a and b:
a = bq + r
Where:
a = dividend
b = divisor
q = quotient
r = remainder
Condition:
0 ≤ r < b
This method is mainly used to find the HCF of two positive integers.
HCF and LCM Relationship
For any two positive integers a and b:
HCF(a, b) × LCM(a, b) = a × b
This formula is useful when one number, HCF, and LCM are given and another number needs to be calculated.
Fundamental Theorem of Arithmetic
Every composite number can be represented as a unique product of prime numbers.
Example:
120 = 2³ × 3 × 5
This concept is commonly used in prime factorization-based MCQs.
Decimal Expansion Rule of Rational Numbers
A rational number in the form p/q has a terminating decimal expansion only when the denominator contains prime factors:
2
5
or both 2 and 5
Examples:
7/20 → Terminating decimal
5/13 → Non-terminating recurring decimal
Remember that non-terminating recurring decimals are rational numbers, not irrational numbers.
Common Mistakes to Avoid While Solving Real Numbers MCQs
Many students understand the concepts of Real Numbers but lose marks because of small mistakes during calculations or while selecting options. While practicing MCQs, pay attention to these common errors:
- Confusing HCF and LCM methods
- Selecting the wrong powers during prime factorization
- Forgetting the condition 0 ≤ r < b in Euclid’s Division Lemma
- Considering non-terminating recurring decimals as irrational numbers
- Making calculation mistakes while applying the HCF × LCM formula
- Solving questions quickly without carefully reading all options
Avoiding these mistakes can improve accuracy and help students solve Class 10 Maths MCQs more confidently.
How to Practice Real Numbers Class 10 MCQs Effectively
Solving questions is useful only when students focus on understanding the concept behind every answer. Follow these simple steps while practicing:
- Revise important formulas before attempting questions
- Understand the logic instead of memorizing answers
- Practice different types of MCQs including formula-based and reasoning questions
- Check explanations after solving each question
- Note difficult questions and revise them regularly before exams
- Regular practice helps improve speed, accuracy, and confidence in solving Real Numbers questions.
