Real Numbers Revision Sheet Class 10 Maths

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Real Numbers — Class 10 Maths Ch 1 Revision Sheet

CBSE 2026–27 · NCERT-aligned · print and revise

State the Fundamental Theorem of Arithmetic and factorise any number into primes
Find HCF and LCM by prime factorisation, and verify the product
Prove √2, √3 and √5 irrational by contradiction
Solve competency-based HCF and LCM word problems

FTA: “One number, one set of primes.”
HCF×LCM: Remember 6 & 20 → 2×60 = 120 = 6×20
Irrational: assume p/q in lowest terms, force a common factor, contradict it.
√ test: non-perfect squares stay irrational (√4 = 2 is rational).

Fundamental Theorem of Arithmetic

n = p₁·p₂…p_k

unique, apart from order

HCF × LCM

HCF(a,b) × LCM(a,b) = a·b

For two numbers only

Irrationality lemma

p | a² ⇒ p | a

p prime

Rational ± irrational

rational + irrational = irrational

e.g., 2 + √3

Concept	Rule / Formula	Example
FTA	unique prime factorisation	96 = 2⁵·3
Prime factor HCF	lowest powers common	HCF 96,404 = 4
Prime LCM	highest powers all	LCM 6,20 = 60
Irrational proof	contradiction; both p,q share a factor	√5 irrational
Product check	HCF × LCM = a × b	4 × 9696 = 96 × 404

Trap	Fix
HCF×LCM for 3 numbers	Only for 2 numbers; for 3 use prime method
√4 → irrational	√4 = 2 rational; check perfect square
Preparing Euclid’s lemma	Removed from the syllabus — use the FTA instead
Weak proof in a 3-mark question	State assumption → contradiction → conclusion

Topper Tip: Always verify HCF×LCM = product as a 10-second sanity check.

Day 2
Re-read vault + factorise 96 and 404

Day 3
Prove √2 and √3 from scratch

Day 5
5 MCQs timed (2 min each)

Day 7
One 3M proof + one CBQ

Q: HCF of 96 & 404?

4 (2²). LCM 9696, product 38784.

Q: Why is √3 irrational?

Assuming √3 = p/q in lowest terms forces 3 to divide both p and q, contradicting “lowest terms”.

Q: Prove 7×11×13+13 is composite?

13(77+1) = 13×78 → more than two factors → composite.

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