What is the Fundamental Theorem of Arithmetic?

The Fundamental Theorem of Arithmetic states that every composite number can be written as a product of prime numbers in exactly one way (unique prime factorisation, ignoring order). Examples: 84 = 2² × 3 × 7; 360 = 2³ × 3² × 5. This uniqueness is why HCF and LCM methods using prime factorisation always give a definite answer.

How do you find HCF and LCM using prime factorisation?

Step 1: Find the prime factorisation of each number. Step 2: For HCF — take the product of the smallest powers of all common prime factors. Step 3: For LCM — take the product of the greatest powers of all prime factors (common and uncommon). Example: HCF(12,18): 12=2²×3, 18=2×3². HCF = 2¹×3¹ = 6. LCM = 2²×3² = 36. Verify: 6×36 = 216 = 12×18 ✓

How do you prove that a number is irrational using the Fundamental Theorem?

The standard proof is by contradiction. To prove √n is irrational (n not a perfect square): Assume √n = p/q in lowest terms (gcd(p,q)=1). Then n = p²/q², so p² = nq². If n has a prime factor k, then k divides p² → k divides p → p = km. Substituting: k²m² = nq² → q² = k²m²/n. For specific n like 2 or 3, this leads to k also dividing q, contradicting gcd(p,q)=1.

Fundamental Theorem of Arithmetic

Real Numbers • Class 10 Mathematics • NCERT • CBSE

The Fundamental Theorem of Arithmetic states: Every composite number can be expressed as a product of primes in exactly one unique way (ignoring order). This theorem underpins HCF, LCM, and proofs that numbers like √2, √3, √5 are irrational.

Key Formulas

Every composite number = unique product of primes
HCF = product of smallest powers of common prime factors
LCM = product of greatest powers of all prime factors
HCF(a,b) × LCM(a,b) = a × b
Terminating decimal ↔ denominator = 2ᵐ × 5ⁿ (no other prime factors)

Frequently Asked Questions

What is the Fundamental Theorem of Arithmetic?: The Fundamental Theorem of Arithmetic states that every composite number can be written as a product of prime numbers in exactly one way (unique prime factorisation, ignoring order). Examples: 84 = 2² × 3 × 7; 360 = 2³ × 3² × 5. This uniqueness is why HCF and LCM methods using prime factorisation always give a definite answer.
How do you find HCF and LCM using prime factorisation?: Step 1: Find the prime factorisation of each number. Step 2: For HCF — take the product of the smallest powers of all common prime factors. Step 3: For LCM — take the product of the greatest powers of all prime factors (common and uncommon). Example: HCF(12,18): 12=2²×3, 18=2×3². HCF = 2¹×3¹ = 6. LCM = 2²×3² = 36. Verify: 6×36 = 216 = 12×18 ✓
How do you prove that a number is irrational using the Fundamental Theorem?: The standard proof is by contradiction. To prove √n is irrational (n not a perfect square): Assume √n = p/q in lowest terms (gcd(p,q)=1). Then n = p²/q², so p² = nq². If n has a prime factor k, then k divides p² → k divides p → p = km. Substituting: k²m² = nq² → q² = k²m²/n. For specific n like 2 or 3, this leads to k also dividing q, contradicting gcd(p,q)=1.

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