Chapter 1 · Question 3

State the Fundamental Theorem of Arithmetic. How does it help in finding HCF and LCM?

Answer Revealed

Direct Answer:

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the prime factors. HCF uses the lowest powers of common primes; LCM uses the highest powers of all primes.

Simple Explanation

Prime factorisation breaks numbers into prime building blocks. Common lowest powers give HCF; all highest powers give LCM.

Exam-Ready Structure

a

and

b

are written as products of prime powers, then HCF is obtained by multiplying the common prime factors with the smaller powers, while LCM is obtained by multiplying every prime factor that occurs with the greatest power. This also gives

\operatorname{HCF}(a,b)\times\operatorname{LCM}(a,b)=a\times b

for two positive integers.

Key Points

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the prime factors.
Use the NCERT formula or theorem carefully.
Write units and final conclusion where applicable.

State the Fundamental Theorem of Arithmetic. How does it help in finding HCF and LCM?