How do you apply Euclid's Division Algorithm to find HCF?

Apply the algorithm repeatedly: Step 1: Write a = bq + r. Step 2: If r = 0, HCF = b. If r ≠ 0, replace a with b, b with r, and repeat. Example: HCF(56, 98): 98=56×1+42 → 56=42×1+14 → 42=14×3+0 → HCF = 14.

How do you determine if a fraction has a terminating or non-terminating decimal?

Express the fraction p/q in lowest terms (cancel all common factors). If the denominator q = 2ᵐ × 5ⁿ (only 2 and 5 as prime factors), it is terminating. Otherwise, non-terminating repeating. Example: 7/20 = 7/(2²×5) → terminating. 1/6 = 1/(2×3) → non-terminating (has factor 3).

State the Fundamental Theorem of Arithmetic.

The Fundamental Theorem of Arithmetic states: Every composite number can be expressed as a product of primes in exactly one way (unique prime factorisation). Example: 84 = 2² × 3 × 7 (unique). This theorem is the basis for finding HCF and LCM using prime factorisation.

Euclid's Division Algorithm

Real Numbers • Class 10 Mathematics • NCERT • CBSE

Euclid's Division Algorithm states: for any two positive integers a and b, there exist unique integers q and r such that a = bq + r (0 ≤ r < b). It is used to find the HCF (Highest Common Factor) of two numbers.

Key Formulas

Euclid's Lemma: a = bq + r (0 ≤ r < b)
HCF × LCM = Product of two numbers
Terminating decimal ↔ denominator = 2ᵐ × 5ⁿ only
For HCF: take smallest powers of common factors
For LCM: take greatest powers of all prime factors

Frequently Asked Questions

How do you apply Euclid's Division Algorithm to find HCF?: Apply the algorithm repeatedly: Step 1: Write a = bq + r. Step 2: If r = 0, HCF = b. If r ≠ 0, replace a with b, b with r, and repeat. Example: HCF(56, 98): 98=56×1+42 → 56=42×1+14 → 42=14×3+0 → HCF = 14.
How do you determine if a fraction has a terminating or non-terminating decimal?: Express the fraction p/q in lowest terms (cancel all common factors). If the denominator q = 2ᵐ × 5ⁿ (only 2 and 5 as prime factors), it is terminating. Otherwise, non-terminating repeating. Example: 7/20 = 7/(2²×5) → terminating. 1/6 = 1/(2×3) → non-terminating (has factor 3).
State the Fundamental Theorem of Arithmetic.: The Fundamental Theorem of Arithmetic states: Every composite number can be expressed as a product of primes in exactly one way (unique prime factorisation). Example: 84 = 2² × 3 × 7 (unique). This theorem is the basis for finding HCF and LCM using prime factorisation.

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