Class 10 Real Numbers Practice Questions
CHAPTER SUMMARY
Class 10 Maths Chapter 1 Real Numbers covers three official syllabus components:
(1) Fundamental Theorem of Arithmetic (unique prime factorization),
(2) HCF & LCM using prime factorization, and
(3) Proofs of irrationality of √2, √3, and √5.
As per CBSE’s updated blueprint, this chapter carries 3–4 marks in the Class 10 Board Exam and often appears as 1-mark MCQs or 2–3 mark application questions. Strong practice in Class 10 Real Numbers Practice Questions helps students build perfect accuracy, eliminate arithmetic mistakes, and improve logical reasoning — essential for scoring 95+ or even 100/100 in board mathematics.
SECTION A — MCQs
- If a number N has the prime factorisation 2⁴ × 3² × 5, which of the following must be a factor of N?
A. 90
B. 120
C. 144
D. 150 Which of the following expressions is always irrational?
A. √2 × 5
B. 2 + √3
C. √9 + √5
D. √3 − √3If the HCF of two numbers is 18 and their LCM is 720, which of the following pairs of numbers is possible?
A. 36 and 360
B. 18 and 720
C. 90 and 144
D. 24 and 540How many factors does the number 2³ × 3⁴ have?
A. 20
B. 24
C. 32
D. 36Which of the following statements is correct?
A. HCF of two numbers can be greater than their LCM
B. Two consecutive natural numbers always have HCF 2
C. LCM of two co-prime numbers is their product
D. √5 + 5 is a rational number
SECTION B — 1-MARK QUESTIONS
Write the prime factorization of 756.
Show that √3 is irrational (short form).
Find the HCF of 108 and 270 using prime factorization.
Determine the LCM of 84 and 210.
State the Fundamental Theorem of Arithmetic.
SECTION C — 2-MARK QUESTIONS
Using prime factorisation, find the HCF and LCM of 144 and 180.
Prove that 2√5 is irrational.
If N = 2⁵ × 3³ × 7, find the number of odd factors of N.
Three bells ring at intervals of 36 minutes, 48 minutes, and 60 minutes. If they start ringing together, after how many minutes will they ring together again?
Show that √2 + √3 is irrational.
SECTION D – 3 MARK QUESTIONS
Find the smallest number which when divided by 24, 36, and 45 leaves the same remainder in each case.
Three ropes are 84 m, 108 m, and 180 m long. Find the greatest length of rope that can be used to cut all the ropes into equal pieces.
If
a = 2⁴ × 3 × 5²
b = 2³ × 3³ × 7
find the HCF and LCM of a and b using prime factorisation.Prove that √6 + √5 is irrational.
A manufacturer cuts metal rods of lengths 480 cm, 600 cm, and 660 cm into equal smaller pieces. Find the maximum possible length of each piece.
SECTION E – 4 MARK QUESTIONS
A company packs pencils in boxes of 50, 75, and 90. Find the minimum number of pencils the company must produce so that all boxes can be filled exactly.
Let N = 2⁶ × 3⁴ × 7².
(a) Find the total number of factors of N.
(b) How many of these factors are multiples of 21?Prove that √2 × √3 × √5 is irrational.
Two cyclists take breaks at intervals of 42 minutes, 63 minutes, and 105 minutes. If they start together at 9:00 am, find all the times before 6:00 pm when they take breaks together.
SECTION F – WORD PROBLEMS
A fruit seller has 240 oranges, 360 apples, and 420 bananas. He wants to make identical fruit baskets using all the fruits.
(a) Find the maximum number of fruits in each basket.
(b) Find the total number of baskets formed.A school wants to arrange 360 chart papers, 432 notebooks, and 504 worksheets into identical bundles without any leftover. Find the maximum number of items in each bundle.
Three traffic signals blink at intervals of 18 seconds, 24 seconds, and 32 seconds. If they start blinking together, after how many seconds will they blink together again?
A marathon track has checkpoints placed at distances of 225 m, 300 m, and 375 m from the starting point. After how many metres will a runner cross all the checkpoints together again?
- A water purification system releases chemicals at intervals of 45 minutes, 60 minutes, and 75 minutes. If the system starts at 8:00 am, list all the times before 6:00 pm when the chemicals are released together.
Class 10 Real Numbers Practice Questions – Complete Chapter Guidance for Board Excellence
The chapter Real Numbers is the foundation of Class 10 Mathematics and plays a crucial role in building number sense, logical reasoning, and accuracy. Solving Class 10 Real Numbers Practice Questions regularly helps students strengthen their understanding of prime factorisation, HCF and LCM, and irrational numbers—concepts that reappear across multiple chapters and competitive-level problems.
According to the latest CBSE syllabus and exam blueprint, Real Numbers carries 3–4 marks in the Class 10 Board Examination, usually in the form of MCQs, short answer questions, and application-based word problems. Though the chapter looks small, the questions are often conceptual and tricky, making consistent practice essential for scoring 95+ or even full marks.
Why Class 10 Real Numbers Practice Questions Are Extremely Important
Many students underestimate this chapter and lose easy marks due to calculation errors or weak conceptual clarity. Practising high-quality Class 10 Real Numbers Practice Questions ensures that students:
Master prime factorisation and apply it accurately
Understand the Fundamental Theorem of Arithmetic deeply
Solve HCF and LCM problems confidently using prime factors
Develop the ability to handle real-life word problems
Prove the irrationality of √2, √3, and √5 logically
Since this chapter does not rely on formulas but on reasoning, repeated practice is the only way to build confidence.
Topics Covered Through These Class 10 Real Numbers Practice Questions
This practice set is strictly aligned with the latest CBSE syllabus and includes questions only from the officially prescribed topics:
Fundamental Theorem of Arithmetic (statement and applications)
Prime factorisation of numbers
Finding HCF and LCM using prime factorisation
Relationship between HCF and LCM
Proofs of irrationality of √2, √3, and √5
Properties of irrational numbers
Application-based problems involving grouping, sharing, and scheduling
By solving these Class 10 Real Numbers Practice Questions, students cover the entire syllabus without wasting time on deleted or irrelevant topics.
Exam Pattern and Question Trends in Real Numbers
In recent CBSE board papers, questions from Real Numbers are often:
Concept-based MCQs testing prime factors
2–3 mark questions on HCF–LCM applications
Proof-based questions on irrational numbers
Word problems involving distribution, grouping, or time intervals
This page includes hard, conceptual, and tricky questions inspired by NCERT, RD Sharma, RS Aggarwal, and previous year question trends, ensuring students are prepared for every possible variation.
How These Practice Questions Help You Score 95+ in Boards
The Class 10 Real Numbers Practice Questions on this page are not random. They are designed to:
Push students beyond rote learning
Improve logical reasoning and number sense
Reduce silly mistakes in factorisation
Strengthen proof-writing skills
Build confidence in solving unfamiliar problems
By regularly solving such questions, students learn how to approach problems step-by-step, which is exactly what examiners expect in board answers.
Common Mistakes Students Make in Real Numbers
Many students lose marks in this chapter due to avoidable errors such as:
Incomplete prime factorisation
Incorrect identification of HCF or LCM
Weak explanation in irrationality proofs
Confusion between HCF-based and LCM-based word problems
Skipping logical steps in reasoning questions
Practising structured and high-level Class 10 Real Numbers Practice Questions helps eliminate these mistakes naturally.
How to Use This Page Effectively
To get the best results from this chapter:
Start with MCQs to revise basic concepts
Move to 1-mark and 2-mark questions for accuracy
Focus on 3-mark and 4-mark questions for reasoning
Spend extra time on word problems, as they are the most scoring
Revise proofs of irrationality multiple times
Using this method, students can convert this chapter into a guaranteed scoring area.
Who Should Use These Class 10 Real Numbers Practice Questions?
This page is ideal for:
CBSE Class 10 students preparing for boards
Students targeting 95–99 marks in Maths
Teachers looking for quality practice material
Students revising before pre-boards and final exams
The questions are challenging enough to stretch top students while still being syllabus-accurate.
Final Words
The chapter Real Numbers may look simple, but it tests fundamental mathematical thinking. With consistent practice using high-quality Class 10 Real Numbers Practice Questions, students can secure full marks from this chapter and build a strong base for advanced topics in Class 10 Maths.
This page is designed to be your one-stop solution for mastering Real Numbers with confidence and clarity.
Supportive Study Material Class 10 Science
CBSE Class 10 Complete Exam Preparation Materials
Best Class 10 Science Worksheets – Complete Practice Sheets for Every Chapter
Table of Contents
Introduction
Benefits of Class 10 Science Worksheets
Physics Worksheets
Chemistry Worksheets
Biology Worksheets
Types of Worksheet Questions
How to Use These Worksheets
Why These Worksheets Improve Scores
Class 10 Science Worksheets
Introduction
The Class 10 Science Worksheets on this page are designed to help students practise every important concept in Physics, Chemistry, and Biology. These worksheets provide clear and structured questions for each chapter so that students can check their understanding, revise faster, and build confidence for exams. Whether preparing for school assessments or the CBSE Class 10 Board Exam, these Class 10 Science Worksheets provide complete support for effective learning.
Students often face difficulty in numericals, diagrams, reasoning questions, and case-based problems. These worksheets make revision easier by offering chapter-wise practice in a simple and organised format.
Benefits of Class 10 Science Worksheets
Solving worksheets regularly helps strengthen concepts and improves your ability to answer different types of questions. These Class 10 Science Worksheets help students:
Revise topics in small, easy steps
Practise numerical problems and diagrams
Improve writing speed and accuracy
Understand reasoning and application-level questions
Prepare for case-based, competency-based, and assertion–reason questions
Identify and fix weak areas through repeated practice
Worksheets make chapter revision quick, simple, and effective.
What You Will Find in These Worksheets
📘 Physics Worksheets
These include numericals, derivations, circuit diagram questions, reasoning questions, and case-based questions from topics like:
Electricity
Magnetic Effects of Electric Current
Light – Reflection and Refraction
Human Eye and the Colourful World
🧪 Chemistry Worksheets
These sheets include chemical equations, reaction identification, table completion, reasoning questions, and numericals from topics like:
Chemical Reactions and Equations
Acids, Bases and Salts
Metals and Non-Metals
Carbon Compounds
🌱 Biology Worksheets
These worksheets include diagram practice, flowcharts, application-based questions, and case studies from topics like:
Life Processes
Control and Coordination
Reproduction
Heredity and Evolution
Our Environment
Types of Worksheet Questions
MCQs (1-mark questions)
Competency-based questions
Passage-based questions
Diagram-based questions
Numerical problems
Short-answer questions
Long-answer questions
Table analysis questions
Reasoning and application questions
These worksheets cover all these question types to ensure complete preparation.
How to Use These Worksheets
Start with chapter notes, then attempt the worksheet.
Solve without looking at the book to check your understanding.
Compare answers with solutions and identify weak areas.
Use the chapter quizzes to quickly revise facts and formulas.
Attempt chapter-wise tests to improve your writing speed.
Finish with sample papers to check your overall preparation.
Following this cycle ensures strong conceptual clarity and full coverage of the syllabus.
Why These Worksheets Improve Scores
These worksheets help you practise every kind of question you may face in the final exam. They improve your time management, increase your confidence, and help you revise concepts repeatedly. Solving them regularly makes Class 10 Science much easier and prepares you well for school exams and the CBSE Board Exam.
Supportive Study Material Class 10 Science
- Class 10 Science Quiz
- Class 10 Science tests
- Class 10 Science Worksheets
- Class 10 Science PYQs
- Class 10 Science NCERT Solution
CBSE Class 10 Complete Exam Preparation Materials
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