Fundamental Theorem of Arithmetic | Class 10 Mathematics

01 · THE THEOREM

Statement of the Theorem

The Fundamental Theorem of Arithmetic states:

Every composite number can be factorised as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

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Canonical Form

For any composite number n, we write:
n = p₁^a₁ × p₂^a₂ × ... × p_k^a_k where p₁ < p₂ < ... < p_k are distinct primes.

02 · HCF & LCM

Finding HCF & LCM using Prime Factorisation

For any two positive integers a and b:

HCF

Highest Common Factor

Product of the smallest power of each common prime factor involved in the numbers.

LCM

Lowest Common Multiple

Product of the greatest power of each prime factor involved in the numbers.

03 · SOLVED EXAMPLE

Step-by-Step Problem Solving

EXAMPLE 1

Find HCF and LCM of 6 and 20 using Prime Factorisation

Step 1: Express numbers as product of primes:

6 = 2¹ × 3¹
20 = 2² × 5¹

Step 2: Calculate HCF (smallest power of common prime factor 2):

HCF(6, 20) = 2¹ = 2

Step 3: Calculate LCM (greatest power of each prime factor 2, 3, 5):

LCM(6, 20) = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60

04 · QUICK SUMMARY

Key Takeaways

Composite numbers have a unique prime factorisation (ignoring order).

HCF uses smallest powers of common prime factors.

LCM uses greatest powers of all prime factors present.