Fundamental Theorem of Arithmetic

• Every composite number can be expressed as a product of primes uniquely (order apart).
• Use prime factorisation to find HCF and LCM efficiently.
• Connect factor trees to NCERT exercises on real numbers.

1. Prime factorisation — break a number into prime factors (e.g. 32760 = 2³ × 3² × 5 × 7 × 13).
2. Fundamental theorem — factorisation is unique.
3. HCF — product of smallest power of each common prime.
4. LCM — product of greatest power of each prime appearing.
5. Application — decide if a rational number has terminating decimal expansion.

HCF and LCM of 96 and 404 by prime factorisation

96 = 2⁵ × 3
404 = 2² × 101
HCF = 2² = 4
LCM = 2⁵ × 3 × 101 = 9696
Check: HCF × LCM = 4 × 9696 = 38784 = 96 × 404 ✓

• Missing a prime factor in the tree.
• Using highest power for HCF instead of lowest common power.
• Thinking 1 is prime (1 is neither prime nor composite).

• Write 140 as product of primes.
• Find HCF(12, 15) and LCM(12, 15).
• If p is prime, how many factors does p² have?

Open the Practice tab for graded questions on Fundamental Theorem of Arithmetic.