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Chapter 1 · Class X Mathematics · MCQ Practice

MCQ Practice Arena

Real Numbers

From Euclid to Irrationals — Build Number-Theory Speed and Accuracy

📋 50 MCQs ⭐ 30 PYQs ⏱ 60 sec/Q

🚀 Start Practising 📖 Revise Notes First

MCQ Bank Snapshot

50Total MCQs

22Easy

20Medium

8Hard

30PYQs

60 secAvg Time/Q

7Topics

Easy 44% Medium 40% Hard 16%

Why Practise These MCQs?

CBSE Class XNTSEState BoardsOlympiad

Real Numbers MCQs blend CBSE Board-style proof recall (Euclid's Lemma, irrationality proofs) with NTSE-style number-theory puzzles. CBSE Boards ask 3–4 marks from this chapter every year; NTSE paper I includes quick HCF/LCM and terminating-decimal classification questions. Most MCQs are solved in under 60 seconds once you recognise the pattern.

Topic-wise MCQ Breakdown

Euclid's Division Lemma & Algorithm10 Q

Fundamental Theorem of Arithmetic9 Q

HCF & LCM via Prime Factorisation10 Q

Irrational Numbers & Proofs8 Q

Terminating vs Non-Terminating Decimals7 Q

Properties of Real Numbers4 Q

Mixed / Conceptual2 Q

Must-Know Formulae Before You Start

Recall these cold before attempting MCQs — they appear in >70% of questions.

$a = bq + r,\ 0 \le r < b\ (\text{Euclid's Lemma})$

$\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$

$\text{Terminating} \Leftrightarrow \text{denominator} = 2^m \cdot 5^n$

$p \mid a^2 \Rightarrow p \mid a\ (\text{prime } p)$

$\sqrt{2},\sqrt{3},\sqrt{5}\ \text{are irrational}$

MCQ Solving Strategy

For Euclid's Algorithm MCQs, just apply a = bq + r repeatedly until remainder = 0. For HCF × LCM MCQs, use the product formula — it's always faster than computing both. Terminating decimal questions: check the denominator in lowest terms; if it only has factors of 2 and 5, it terminates. For irrationality proofs in MCQ form, watch for the "assume rational, derive contradiction" logic.

⚠ Common Traps & Errors

⚠HCF × LCM = a × b works only for TWO numbers, not three
⚠LCM is always ≥ HCF — never confuse them
⚠Remainder in Euclid's Lemma must satisfy 0 ≤ r < b, not 0 < r ≤ b
⚠p | a² ⟹ p | a is true ONLY when p is prime
⚠2/6 is NOT in lowest terms — simplify before checking denominator

Difficulty Ladder

Work through each rung in order — do not jump to Hard before mastering Easy.

① Easy

Find HCF/LCM by prime factorisation, apply Euclid's algorithm to small numbers

② Medium

Irrationality proofs (MCQ), terminating decimal classification, HCF×LCM product

③ Hard

Mixed problems: combine Euclid's lemma with FTA; three-number LCM

★ PYQ

CBSE — prove √p irrational; NTSE — direct HCF/LCM application

Continue Your Preparation

📖 Chapter Notes Revise concepts & theory 🧠 MCQ Practice Topic-wise Questions 📘✔️ Exercises Step-by-step NCERT solutions ✔️❌ True / False Concept-based True & False ➡ Next Chapter Polynomials

🎯 Knowledge Check

Maths — REAL NUMBERS

50 Questions Class 10 MCQs

The least positive integer that is divisible by 12, 18, and 21 is:

a a. 72 b b. 252 c c. 126 d d. 84

The HCF of 65 and 117 is:

a a. 1 b b. 13 c c. 5 d d. 17

The largest number which divides 245 and 1029 exactly is:

a a. 7 b b. 49 c c. 35 d d. 14

Which of the following is irrational?

a a. $\frac{22}{7}$ b b. 0.16 c c. 0.125 d d. $\sqrt{3}$

Euclid’s division lemma applies to:

a a. Only natural numbers b b. All positive integers c c. Prime numbers only d d. All real numbers

The decimal form of a rational number is:

a a. Terminating b b. Non-terminating but repeating c c. Either (A) or (B) d d. Always non-terminating non-repeating

The product of a rational and an irrational number is:

a a. Always irrational b b. Sometimes rational and sometimes irrational c c. Always rational d d. Always integer

“Every composite number can be expressed (factorised) as a product of primes,” is called:

a a. Prime Fundamental Law b b. Fundamental Theorem of Arithmetic c c. Factorisation Law d d. Euclid’s Lemma

The LCM and HCF of two numbers are 180 and 6 respectively. If one number is 30, the other number is:

a a. 24 b b. 36 c c. 42 d d. 36

The number 0.101001000100001...0.101001000100001...0.101001000100001... is:

a a. Rational b b. Terminating c c. Irrational d d. Integer

If p and q are coprime numbers, HCF(pq, p + q) is:

a a. 1 b b. $pq$ c c. 1 d d. $p + q$

If the HCF of 65 and 117 can be written as 65x + 117y, then the value of x is:

a a. 5 b b. 3 c c. 2 d d. -9

The decimal expansion of $\frac{23}{16}$ is:

a a. Terminating b b. Non-terminating, repeating c c. Terminating d d. Non-terminating, non-repeating

The number $\sqrt{2}$ is:

a a. Rational b b. Irrational c c. Integer d d. Natural number

The LCM of 1 and any number aaa is:

a a. 0 b b. a c c. 1 d d. $ a^2$

If a whole number has a non-terminating, non-repeating decimal expansion, then it is:

a a. Integer b b. Rational c c. Irrational d d. Real

The smallest prime number is:

a a. 1 b b. 2 c c. 2 d d. 3

Every rational number can be written as:

a a. In the form $m/n$ where $m,~n$ are integers and $ eq 0$ b b. In the form $p/q$ with denominator $q>0$and $p,~q$ integers c c. Both (A) and (B) d d. None of these

Which of the following has a terminating decimal expansion?

a a. $\frac{3}{7}$ b b. $\frac{5}{8}$ c c. $\frac{2}{9}$ d d. $\frac{4}{11}$

The Fundamental Theorem of Arithmetic is related to:

a a. Division of integers b b. Unique factorization of natural numbers c c. Multiplication of numbers d d. None of these

The square root of a non-perfect square natural number is:

a a. Rational b b. Irrational c c. Integer d d. Prime

The decimal expansion of $\frac{77}{600}$ will terminate after:

a a. 1 decimal place b b. 2 decimal places c c. 3 decimal places d d. 4 decimal places

If $n=pq $ where $p,~q$ are prime, then how many factors does $n $have?

a a. 1 b b. 2 c c. 3 d d. 4

The sum of two irrational numbers is always:

a a. Rational b b. Irrational c c. Sometimes rational, sometimes irrational d d. Integer

The decimal expansion of $\frac{17}{40}$ is:

a a. Terminating b b. Non-terminating repeating c c. Terminating d d. Non-terminating non-repeating

The sum of a rational and an irrational number is:

a a. Always rational b b. Always irrational c c. Always integer d d. Always prime

HCF(84, 90) = ?

a a. 2 b b. 6 c c. 12 d d. 30

There are infinitely many primes divisible by:

a a. 2 b b. 5 c c. 3 d d. None of these (no prime is divisible by another prime except itself)

The number $\frac{2}{3}$ is:

a a. Irrational b b. Rational c c. Integer d d. Natural

If a number's decimal representation terminates, then its denominator in lowest form is:

a a. Product of 2’s only b b. Product of 5’s only c c. Product of 2’s and/or 5’s d d. Product of any primes

Euclid’s algorithm is used to find:

a a. LCM b b. Decimal expansion c c. Prime factorization d d. HCF

$\frac{13}{50}$ in decimal form is:

a a. 0.26 b b. 0.6 c c. 0.236 d d. 0.26

Which of the following is not rational?

a a. 0 b b. 2.73 c c. $\frac{7}{8}$ d d. $\sqrt{7}$

The decimal expansion of $\frac{4}{11}$ is:

a a. Terminating b b. $0.overline{36}$ c c. 0.172839... d d. Irrational

The HCF of 15, 25, and 30 is:

a a. 1 b b. 5 c c. 5 d d. 15

LCM (16, 24) is:

a a. 16 b b. 32 c c. 48 d d. 48

Product of two non-zero rational numbers is:

a a. Always integer b b. Always irrational c c. 0 d d. Rational

The number $\sqrt{49}$ is:

a a. 6 b b. 5 c c. 8 d d. 7

The largest 2-digit prime number is:

a a. 97 b b. 99 c c. 97 d d. 89

If a number is not rational, it is:

a a. Integer b b. Prime c c. Irrational d d. Odd

The HCF of two consecutive even numbers is always:

a a. 2 b b. 2 c c. 1 d d. 0

The number 17 is:

a a. Even b b. Composite c c. Irrational d d. Prime

An irrational number between 2 and 3 is:

a a. 2.1 b b. $\sqrt{5}$ c c. 2.9 d d. 3

For any two positive integers a and b, a = bq + r. 0 ? r < b. Here, r is:

a a. Quotient b b. Product c c. Remainder d d. Factor

Which of the following is a non-terminating non-repeating decimal?

a a. 0.333... b b. 0.424242... c c. 0.25 d d. 0.101001000100001...

LCM of 9 and 15 is:

a a. 30 b b. 35 c c. 45 d d. 45

The denominator of a rational number with terminating decimal expansion has the form:

a a. $2^n$ only b b. $5^m$ only c c. $2^n5^m$ , where $n, ~m$ are non-negative integers d d. $7^k$

The sum of two rational numbers is:

a a. Always irrational b b. Always rational c c. Always integer d d. Sometimes irrational

Which of these is not a real number?

a a. -7 b b. 0 c c. 5 d d. $\sqrt{-3}$

The set of real numbers consists of:

a a. Only rationals b b. Only irrationals c c. Rationals and integers d d. All rational and irrational numbers

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Frequently Asked Questions

Real numbers include all rational and irrational numbers, representing all points on the number line.

Rational numbers can be expressed asp/qp/qp/qwherepppandqqqare integers andq?0q \neq 0q?=0.

Irrational numbers cannot be expressed asp/qp/qp/q; their decimal expansion is non-terminating and non-repeating.

Euclid, an ancient Greek mathematician, proposed the division lemma used for finding HCF.

For any two positive integersaaaandbbb, there exist unique integersqqqandrrrsuch thata=bq+ra = bq + ra=bq+r, where0=r<b0 \leq r < b0=r<b.

It helps find the Highest Common Factor (HCF) of two numbers using repeated division.

It is the process of applying Euclid’s Lemma repeatedly to find the HCF of two numbers.

HCF (Highest Common Factor) is the greatest number that divides two or more numbers exactly.

LCM (Least Common Multiple) is the smallest number divisible by the given numbers.

HCF×LCM=Product of the two numbers\text{HCF} \times \text{LCM} = \text{Product of the two numbers}HCF×LCM=Product of the two numbers.

Prime numbers are natural numbers greater than 1 that have only two factors: 1 and itself.

Composite numbers have more than two factors. Examples: 4, 6, 8, 9.

Every composite number can be expressed as a product of primes in a unique way, except for order of factors.

Expressing a number as a product of prime numbers.

List prime factors of each number and multiply common factors with least power.

REAL NUMBERS – Learning Resources

Detailed Notes

True / False

NCERT Solutions

ACADEMIA AETERNUM तमसो मा ज्योतिर्गमय · Est. 2025

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Real Numbers

MCQ Bank Snapshot

Why Practise These MCQs?

Topic-wise MCQ Breakdown

Must-Know Formulae Before You Start

MCQ Solving Strategy

⚠ Common Traps & Errors

Difficulty Ladder

Continue Your Preparation

Maths — REAL NUMBERS

Frequently Asked Questions

Q 01 What are real numbers?

Q 02 Define rational numbers.

Q 03 What are irrational numbers?

Q 04 Who gave the Euclid’s Division Lemma?

Q 05 State Euclid’s Division Lemma.

Q 06 What is the purpose of Euclid’s Division Lemma?

Q 07 What is the Euclid’s Division Algorithm?

Q 08 What is HCF?

Q 09 What is LCM?

Q 10 State the relation between HCF and LCM.

Q 11 What are prime numbers?

Q 12 What are composite numbers?

Q 13 What is the Fundamental Theorem of Arithmetic?

Q 14 What is prime factorization?

Q 15 How do you find the HCF using prime factorization?

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REAL NUMBERS – Learning Resources

Detailed Notes

True / False

NCERT Solutions

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