Is it correct to say: Using HCF × LCM = product for three numbers?

The rule holds for two numbers only. For three numbers, find the HCF and LCM separately from the prime factorisations.

Is it correct to say: Taking the greatest powers for HCF and the smallest powers for LCM?

HCF = smallest power of each COMMON prime; LCM = greatest power of EVERY prime involved. Check: the HCF must divide both numbers, and the LCM must be a multiple of both.

Is it correct to say: Not stating that a and b are co-prime at the start of an irrationality proof?

The contradiction comes from that assumption. Write 'a and b are co-prime (no common factor other than 1), b ≠ 0' in the first line.

Is it correct to say: Saying 'a number ends in 0 because it is even'?

A number ends in 0 only if its prime factorisation has BOTH 2 and 5. 6ⁿ = 2ⁿ × 3ⁿ is even but never ends in 0.

Is it correct to say: Assuming the sum of two irrational numbers is always irrational?

√2 + (−√2) = 0 is rational. Only rational ± irrational is guaranteed to be irrational.

Is it correct to say: Treating 1 as a prime in a factorisation?

1 is neither prime nor composite. Never write 1 as a factor in a prime factorisation.

📄 Formula sheet (PDF)

Real Numbers

Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, HCF × LCM = product, and proofs of irrationality — NCERT Class 10 Maths Ch 1

📐 12 formulas✏️ 4 examples🎯 6 practice⚖️ Unit I (Number Systems): 6 marks🏫 CBSE📚 Class 10✓ 2025–26 syllabus

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Board Exam Tips

→Write every prime factorisation in exponent form with primes in increasing order: 3150 = 2 × 3² × 5² × 7. It makes HCF and LCM a matter of reading off powers.
→HCF → smallest power of each COMMON prime. LCM → greatest power of EVERY prime that appears. Do not swap them.
→HCF × LCM = product works for TWO numbers only. Never use it for three numbers.
→Irrationality proofs: begin with 'Let us assume, to the contrary, that … is rational', take a and b co-prime, and end with 'This contradicts … so our assumption is wrong'.
→For numbers like 5 + 2√3, assume it is rational, isolate √3 on one side, and quote that √3 is irrational. Two or three clean lines are enough.
→Word problems: 'together again / at the same time' needs the LCM; 'largest size / greatest number that divides' needs the HCF.

📐 Formulas(12)

Fundamental Theorem of Arithmetic★ Board fav

n = p_{1}^{a_{1}} p_{2}^{a_{2}} \dots p_{k}^{a_{k}}, p_{1} < p_{2} < \dots < p_{k} primes

Symbol	Meaning
$n$	Composite natural number being factorised
$p_{1}, p_{2}, \dots, p_{k}$	Distinct prime factors of n, in increasing order
$a_{1}, a_{2}, \dots, a_{k}$	Powers (natural numbers) of the primes

HCF by Prime Factorisation

HCF = product of smallest powers of common primes

LCM by Prime Factorisation

LCM = product of greatest powers of all primes involved

HCF × LCM = Product of Two Numbers★ Board fav

HCF (a, b) \times LCM (a, b) = a \times b

Symbol	Meaning
$a, b$	Two positive integers

LCM from HCF

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

Co-prime Numbers

HCF (a, b) = 1 \Rightarrow LCM (a, b) = a \times b

HCF Always Divides LCM

HCF (a, b) ∣ LCM (a, b)

Three Numbers: Product Rule Fails

HCF (p, q, r) \times LCM (p, q, r) \neq = p \times q \times r in general

When Does a Number End in 0?

n ends in 0 ⟺ 2 ∣ n and 5 ∣ n (10 = 2 \times 5)

Prime Dividing a Square

p prime, p ∣ a^{2} \Rightarrow p ∣ a

Symbol	Meaning
$p$	A prime number
$a$	A positive integer

√2 Is Irrational (Proof by Contradiction)★ Board fav

2 = \frac{a}{b}, HCF (a, b) = 1 \Rightarrow a^{2} = 2 b^{2} \Rightarrow 2 ∣ a and 2 ∣ b

Rational and Irrational Combined★ Board fav

r \in Q, s \in / Q \Rightarrow r + s, r - s \in / Q; r s \in / Q (r \neq = 0)

Symbol	Meaning
$r$	A rational number
$s$	An irrational number

✏️ Solved Examples

1Solved Exampleeasy4 steps

Find the HCF and LCM of 84 and 120 by prime factorisation, and verify that HCF × LCM = product of the numbers.

Write the prime factorisations.

84 = 2^{2} \times 3 \times 7, 120 = 2^{3} \times 3 \times 5

2Solved Exampleboard3 steps

Check whether 12ⁿ can end with the digit 0 for any natural number n.

A number ends in 0 only if it is divisible by 10 = 2 × 5, so its prime factorisation must contain both 2 and 5.

3Solved Exampleboard5 steps

Prove that √3 is irrational.

Assume, to the contrary, that √3 is rational. Then we can find integers a and b (b ≠ 0) with no common factor other than 1 such that

3 = \frac{a}{b}

4Solved ExampleHOTS4 steps

Given that √3 is irrational, prove that 5 + 2√3 is irrational.

Assume, to the contrary, that 5 + 2√3 is rational. Then there exist co-prime integers a and b (b ≠ 0) such that

5 + 23 = \frac{a}{b}

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Using HCF × LCM = product for three numbers
✓The rule holds for two numbers only. For three numbers, find the HCF and LCM separately from the prime factorisations.
2
Taking the greatest powers for HCF and the smallest powers for LCM
✓HCF = smallest power of each COMMON prime; LCM = greatest power of EVERY prime involved. Check: the HCF must divide both numbers, and the LCM must be a multiple of both.
3
Not stating that a and b are co-prime at the start of an irrationality proof
✓The contradiction comes from that assumption. Write 'a and b are co-prime (no common factor other than 1), b ≠ 0' in the first line.
4
Saying 'a number ends in 0 because it is even'
✓A number ends in 0 only if its prime factorisation has BOTH 2 and 5. 6ⁿ = 2ⁿ × 3ⁿ is even but never ends in 0.
5
Assuming the sum of two irrational numbers is always irrational
✓√2 + (−√2) = 0 is rational. Only rational ± irrational is guaranteed to be irrational.
6
Treating 1 as a prime in a factorisation
✓1 is neither prime nor composite. Never write 1 as a factor in a prime factorisation.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Express 3150 as a product of its prime factors.
Q2
The HCF of two numbers is 18 and their product is 12 960. Find their LCM.
Q3
Can two numbers have 16 as their HCF and 380 as their LCM? Give a reason.
Q4
Three traffic lights change every 48 s, 72 s and 108 s. They change together at 7:00:00 a.m. When will they next change together?
Q5
Explain why 5 × 7 × 11 + 7 is a composite number.
Q6
Given that √2 is irrational, prove that 4 − 3√2 is irrational.

📝 Notes

Real Numbers

This chapter rests on one idea: every composite number breaks into primes in exactly one way. Once you accept that, HCF, LCM and irrationality proofs all follow.

From factorisation to HCF and LCM

Write both numbers as products of prime powers. The HCF uses only the primes the numbers share, each at its smaller power. The LCM uses every prime that appears, each at its larger power. Multiplying them gives back the product of the two numbers, because for each prime you have taken the smaller power once and the larger power once:

HCF(a, b) × LCM(a, b) = a × b.

This works for two numbers only. With three numbers, always go back to the factorisations.

Uniqueness answers "can it ever …?" questions

Questions like "Can 4ⁿ end in 0?" are really about uniqueness. A number ending in 0 is divisible by 10 = 2 × 5. Since the prime factorisation of 4ⁿ is 2²ⁿ, no 5 can ever appear, so the answer is no for every n.

Irrationality proofs: one pattern

Every irrationality proof in this chapter is a proof by contradiction:

Assume √p = a/b with a, b co-prime and b ≠ 0.
Square: p b² = a², so p divides a², and therefore p divides a.
Put a = pc to get b² = p c², so p divides b too.
p divides both a and b, which contradicts co-primeness.

For expressions such as 5 + 2√3 or 4 − 3√2, you do not need to repeat this. Assume the expression is rational, rearrange to get √3 (or √2) equal to a fraction of integers, and point out the contradiction.

Quick checks

The HCF divides each number; each number divides the LCM.
HCF ≤ smaller number, and LCM ≥ larger number.
Rational ± irrational is irrational. Non-zero rational × irrational is irrational.