Q1. Find the HCF of 96 and 404 by Euclid's algorithm.
404 = 96 × 4 + 20
96 = 20 × 4 + 16
20 = 16 × 1 + 4
16 = 4 × 4 + 0 ⇒ HCF = 4
Real Numbers — Class 10
Every number breaks into primes in exactly one way. Use the HCF & LCM calculator and master the proofs the board loves.
Euclid's division lemmaFor positive integers a and b, there exist unique q and r with a = bq + r, 0 ≤ r < b. Repeating it gives the HCF.
Fundamental theorem of arithmeticEvery composite number can be written as a product of primes in only one way (apart from order).
HCF & LCM relationFor two numbers, HCF × LCM = product of the numbers. Use this to find one from the other.
Rational & irrationalA rational number p/q (q≠0) has a terminating or repeating decimal; irrationals (like √2, π) do not.
HCF & LCM calculator Interactive
Enter two or three numbers to get their HCF and LCM, with the prime factorisation.
HCF—
LCM—
Numericals practice Interactive
HCF, LCM and product problems with steps.
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Solved examples
Q2. Prove √2 is irrational (idea).
Assume √2 = p/q in lowest terms.
Then 2q² = p², so p is even ⇒ p = 2k ⇒ 2q² = 4k² ⇒ q² = 2k², so q is even too.
Both even contradicts "lowest terms" ⇒ √2 is irrational.
Formula sheet
| Quantity | Formula |
|---|---|
| Euclid's lemma | a = bq + r, 0 ≤ r < b |
| HCF × LCM | HCF(a,b) × LCM(a,b) = a × b |
| Terminating decimal | p/q terminates ⇔ q = 2ᵐ·5ⁿ |
Common mistakes & exam wins
- HCF × LCM = product works only for TWO numbers, not three.
- A fraction p/q gives a terminating decimal only if q has just 2s and 5s as prime factors.
- Euclid's algorithm: keep replacing the pair (a, b) with (b, remainder) until the remainder is 0.
- The HCF is the SMALLEST combination; the LCM is the product of the highest powers of all primes.
Frequently asked questions
What is the fundamental theorem of arithmetic?
Every composite number can be expressed as a product of primes, and this factorisation is unique except for the order of the factors.
What is the relation between HCF and LCM?
For any two positive integers, HCF × LCM equals the product of the numbers.
How do you know a decimal will terminate?
A rational number p/q (in lowest terms) has a terminating decimal exactly when q is of the form 2ᵐ × 5ⁿ.