Class 10 · Chapter 1 · Number Systems unit (6 of 80 marks)

Real Numbers Class 10: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 40 board-style questions.

Practise this chapter Join freeFree account, no card: +5 practice questions on every chapter, with instant feedback.

40 questions
12 multiple choice, 3 assertion–reason, 9 very short answer, 8 short answer, 5 long answer, 3 case study
About 8 hours to master

Number Systems unit: 6 of 80 theory marks (Real Numbers is the only chapter in this unit).

NCERT textbook: Chapter 1 on ncert.nic.in (PDF)

Revision notes

Real Numbers — revision notes

1. The Fundamental Theorem of Arithmetic (FTA)

Every composite number can be written as a product of primes, and this factorisation is unique apart from the order of the factors. Write it in index form, e.g. $3960 = 2^{3} \times 3^{2} \times 5 \times 11$ .

Find factors by repeated division by the smallest prime (2, 3, 5, 7, 11, …) or with a factor tree.
Uniqueness lets you compare exponents: if $2^{x} 3^{y} = 2^{4} 3^{3}$ then $x = 4, y = 3$ .
A number ends in $0$ only if its prime factorisation contains both $2$ and $5$ ; it ends in $5$ only if it contains $5$ (and no $2$ ).

2. HCF and LCM by prime factorisation

HCF = product of the smallest power of each common prime factor.
LCM = product of the greatest power of every prime factor involved.
For two positive integers only: $HCF (a, b) \times LCM (a, b) = a \times b$ .
HCF always divides LCM, so e.g. HCF $14$ , LCM $204$ is impossible.

3. Word problems — which one?

HCF: “greatest/largest”, splitting into equal groups, arranging in rows/stacks/trays, cutting into longest equal pieces. “Leaves remainder $r$ ” → take HCF of (numbers $- r$ ).
LCM: “least/smallest”, events happening together again (bells, lights, laps). “Leaves remainder $r$ in each case” → LCM $+ r$ .

4. Irrational numbers

Key result: if a prime $p$ divides $a^{2}$ , then $p$ divides $a$ . Use it to prove $\sqrt{2}, \sqrt{3}, \sqrt{5}$ irrational by contradiction.

rational $\pm$ irrational = irrational; non-zero rational $\times$ or $\div$ irrational = irrational.
irrational $\pm$ irrational may be rational: $(2 + \sqrt{3}) + (2 - \sqrt{3}) = 4$ .

Worked example 1

Find HCF and LCM of $96$ and $360$ .
$96 = 2^{5} \times 3$ , $360 = 2^{3} \times 3^{2} \times 5$ . HCF $= 2^{3} \times 3 = 24$ ; LCM $= 2^{5} \times 3^{2} \times 5 = 1440$ . Check: $24 \times 1440 = 34560 = 96 \times 360$ .

Worked example 2

Prove that $\sqrt{5}$ is irrational.
Suppose $\sqrt{5} = a / b$ , $a, b$ coprime. Then $a^{2} = 5 b^{2}$ , so $5 ∣ a^{2}$ and hence $5 ∣ a$ . Put $a = 5 c$ : $25 c^{2} = 5 b^{2} \Rightarrow b^{2} = 5 c^{2}$ , so $5 ∣ b$ . Then $5$ divides both $a$ and $b$ — contradiction. Hence $\sqrt{5}$ is irrational.

Worked example 3

Show that $7 + 2 \sqrt{3}$ is irrational (given $\sqrt{3}$ irrational).
If $7 + 2 \sqrt{3} = r$ (rational), then $\sqrt{3} = \frac{r - 7}{2}$ , which is rational — contradiction.

Common errors

Using $HCF \times LCM = product$ for three numbers.
Taking the highest power for HCF (or the lowest for LCM).
In irrationality proofs, forgetting to state that $a$ and $b$ are coprime, or not stating the contradiction clearly.
Leaving a “prime factorisation” with a composite factor such as $77$ or $143$ .

Board-exam tips

Always show the prime factorisation — it carries a method mark even if the final answer slips.
In word problems write a final sentence with units (“216 books per stack”, “9:00 a.m.”).
Irrationality proofs are regular 2–3 mark items: learn the structure assume → square → prime divides → contradiction.

Topics in this chapter: Fundamental Theorem of Arithmetic · HCF and LCM by prime factorisation · Irrational numbers · Applications of HCF and LCM.

Formulas for this chapter: Chapter 1 on the Class 10 formula sheet (free PDF and A3 poster).

Real Numbers Class 10: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 171 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceHCF and LCM by prime factorisation

Given that $HCF (96, 156) = 12$ , the value of $LCM (96, 156)$ is

Show worked answer

$\begin{aligned} LCM & = \frac{96 \times 156}{12} \\ = 8 \times 156 \\ = 1248 \end{aligned}$ A1
Correct option: (a) $1248$

Q2·1 mark·Multiple choiceIrrational numbers

Which one of the following numbers is irrational?

Q3·1 mark·Assertion–reasonFundamental Theorem of Arithmetic

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): For no natural number $n$ does $15^{n}$ end with the digit $0$ .

Reason (R): A natural number ends with the digit $0$ only if its prime factorisation contains both $2$ and $5$ .

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerHCF and LCM by prime factorisation

Using prime factorisation, find the HCF and the LCM of $144$ and $198$ .

Show worked answer

Answer: HCF

= 18

, LCM

= 1584

$144 = 2^{4} \times 3^{2}$ and $198 = 2 \times 3^{2} \times 11$ M1
HCF $= 2 \times 3^{2} = 18$ ; LCM $= 2^{4} \times 3^{2} \times 11 = 1584$ A1

Q5·2 marks·Short answerIrrational numbers

It is given that $\sqrt{5}$ is irrational. Prove that $2 \sqrt{5} - 3$ is not a rational number.

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerApplications of HCF and LCM

Two tankers hold $850$ litres and $680$ litres of milk. Find the greatest capacity of a container that can measure out the milk of each tanker an exact number of times. How many times in all is the container filled?

Show worked answer

Answer:

170

litres;

9

times

$850 = 2 \times 5^{2} \times 17$ and $680 = 2^{3} \times 5 \times 17$ M1
Greatest capacity $=$ HCF $= 2 \times 5 \times 17 = 170$ litres A1
Number of fillings $\begin{aligned} = \frac{850}{170} + \frac{680}{170} & = 5 + 4 \\ = 9 \end{aligned}$ A1

Q7·3 marks·Short answerApplications of HCF and LCM

What is the smallest number above $1000$ that leaves no remainder when divided by $15$ , by $24$ and by $36$ ?

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerApplications of HCF and LCM

A relief camp has $336$ notebooks, $240$ pens and $192$ erasers. They are packed into identical kits, every item used, with the greatest possible number of kits. (a) How many kits are made, and what does each kit contain? (b) Delivery vans for the kits leave the camp every $20$ , $30$ and $45$ minutes on three routes. If all three leave together at $8$ a.m., when do they next leave together?

Show worked answer

Answer: (a)

48

kits of

7

notebooks,

5

pens,

4

erasers; (b)

11

a.m.

(a) $336 = 2^{4} \times 3 \times 7$ , $240 = 2^{4} \times 3 \times 5$ , $192 = 2^{6} \times 3$ M1
Number of kits $=$ HCF $= 2^{4} \times 3 = 48$ A1
Each kit: $\frac{336}{48} = 7$ notebooks, $\frac{240}{48} = 5$ pens, $\frac{192}{48} = 4$ erasers A1
(b) $20 = 2^{2} \times 5$ , $30 = 2 \times 3 \times 5$ , $45 = 3^{2} \times 5$ ; LCM $= 2^{2} \times 3^{2} \times 5 = 180$ minutes M1
$180$ minutes $= 3$ hours, so they next leave together at $11$ a.m. A1

Q9·5 marks·Long answerIrrational numbers

Show that $\sqrt{5}$ cannot be written as $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ . Use this to show that $3 + \sqrt{20}$ is also irrational.

Section E style · Case study (4 marks)

Q10·4 marks·Case studyApplications of HCF and LCM

Tiling a hall. A rectangular hall floor measures $1020$ cm by $720$ cm. It is to be covered with identical square tiles, as large as possible, with no tile cut.

Write $720$ as a product of its prime factors. [1 mark]
Find the side of the largest square tile that can be used. [1 mark]
How many such tiles are needed to cover the floor? [2 marks]

Show worked answer

$720 = 2^{4} \times 3^{2} \times 5$ A1
$1020 = 2^{2} \times 3 \times 5 \times 17$ ; HCF $= 2^{2} \times 3 \times 5 = 60$ cm A1
Along the length $\frac{1020}{60} = 17$ tiles; along the breadth $\frac{720}{60} = 12$ tiles M1
Number of tiles $= 17 \times 12 = 204$ A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 10 assertion–reason questions · the Class 10 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can write any number as a product of primes and find the HCF and LCM of two numbers quickly and accurately.

Read first: 1. The Fundamental Theorem of Arithmetic (FTA); 2. HCF and LCM by prime factorisation 8 practice questions · checkpoint: 3 questions, 5 marks, pass 80%

Practise step 1

Step 2

Board standard

You can handle every standard board question: HCF/LCM word problems, the HCF × LCM rule, and the short 'show that it is irrational' proofs.

Read first: 2. HCF and LCM by prime factorisation; 3. Word problems — which one?; 4. Irrational numbers; Worked examples 1-3 18 practice questions · checkpoint: 4 questions, 12 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write complete 5-mark and case-study answers, with the prime factorisation shown, a clear final sentence and units, so no step mark slips away.

Read first: 3. Word problems — which one?; 4. Irrational numbers; Board-exam tips 7 practice questions · checkpoint: 3 questions, 14 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can crack the trickiest questions: finding unknown powers, listing all possible number pairs and spotting which sums and products of surds stay irrational.

Read first: 4. Irrational numbers; Common errors 7 practice questions · checkpoint: 3 questions, 8 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 6 of the 40 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceFundamental Theorem of Arithmetic

The prime factorisation of $1386$ is

(a) $2 \times 3^{2} \times 7 \times 11$
(b) $2^{2} \times 3 \times 7 \times 11$
(c) $2 \times 3 \times 7^{2} \times 11$
(d) $2 \times 3^{2} \times 77$

Q2·1 mark·Multiple choiceFundamental Theorem of Arithmetic

For some natural number $n$ , which of the following numbers can end with the digit $5$ ?

(a) $12^{n}$
(b) $14^{n}$
(c) $35^{n}$
(d) $18^{n}$

Q3·2 marks·Very short answerFundamental Theorem of Arithmetic

Express $5005$ as a product of its prime factors.

Worked solution·2 marks·Very short answerIrrational numbers

Given that $\sqrt{7}$ is irrational, show that $4 - 3 \sqrt{7}$ is irrational.

Full worked solution Free, no sign-in

Suppose $4 - 3 \sqrt{7} = r$ , a rational number. Then $\sqrt{7} = \frac{4 - r}{3}$ . M1
$\frac{4 - r}{3}$ is rational, so $\sqrt{7}$ would be rational — a contradiction. Hence $4 - 3 \sqrt{7}$ is irrational. A1

Every question here has a full worked solution. See them all →

Stretch yourself: original number theory problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I61 · Problem I03 · Problem I43 · Problem I50. All 32 →

Themed lesson: Children’s Day (India) maths lesson · Fibonacci Day maths lesson · National Mathematics Day maths lesson (starter, activity and worked answers). All themed maths

Knowledge organiser: Real Numbers on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Real Numbers

Leaving a composite factor such as 77 or 143 in a 'prime factorisation' loses the A1 mark. Fix: keep dividing until every factor is prime, and write the answer in index form.
Using HCF × LCM = product for three numbers. Fix: the rule holds for two numbers only; for three numbers find HCF and LCM separately from the factorisations.
Mixing up the powers: taking the highest power for HCF or the lowest for LCM. Fix: say it aloud before writing: HCF = smallest power of common primes, LCM = greatest power of every prime.
Irrationality proof without 'a and b are coprime' or without a clearly stated contradiction loses the reasoning marks. Fix: always write assume a/b in lowest terms → square → p divides a and b → contradiction → hence irrational.
In '7 + 2√3 is irrational' questions, forgetting to say 'since √3 is irrational (given)' at the contradiction step. Fix: name the known irrational fact explicitly in the last line.
Word problems answered with a bare number. Fix: decide HCF (equal groups, greatest) or LCM (together again, least) in writing, then finish with a sentence and units, e.g. '9:00 a.m.' or '216 books per stack'.
Remainder problems: forgetting to add back (LCM + r) or subtract first (HCF of numbers − r). Fix: write the adjustment step as its own line.

Exam strategy: how answers are step-marked · the proofs, written out · competency-based questions

Examiner Insights: common mistakes in CBSE Class 10 Maths, with fixes (our analysis of public sources) →

Real Numbers in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 1, 21, 26
Sample paper 2 (Board standard): questions 19, 21, 26
Sample paper 3 (Above board): questions 1, 21, 26
Sample paper 4 (Above board): questions 1, 21, 26
Sample paper 5 (95+ challenge): questions 1, 21, 26

Want it against the clock? Take a timed 30-mark chapter test on Real Numbers, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Real Numbers: common questions

What is in CBSE Class 10 Maths Chapter 1, Real Numbers?

The topics are: Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, Irrational numbers, Applications of HCF and LCM. This page has revision notes for each, and 40 board-style practice questions with step-marked solutions.

How many marks is Real Numbers worth?

It is in the Number Systems unit, which carries 6 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Real Numbers?

Leaving a composite factor such as 77 or 143 in a 'prime factorisation' loses the A1 mark. Fix: keep dividing until every factor is prime, and write the answer in index form. Using HCF × LCM = product for three numbers. Fix: the rule holds for two numbers only; for three numbers find HCF and LCM separately from the factorisations.

Are the Real Numbers important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Real Numbers — revision notes

1. The Fundamental Theorem of Arithmetic (FTA)

2. HCF and LCM by prime factorisation

3. Word problems — which one?

4. Irrational numbers

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Real Numbers Class 10: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Real Numbers

Real Numbers in our sample papers

Real Numbers: common questions

Work through the Route to 95 with your progress saved