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The Fundamental Theorem of Arithmetic

Every composite number breaks into primes in exactly one way — and that single idea lets you find HCF and LCM quickly.

About 9 min readReviewed by a CBSE Mathematics teacherSyllabus 2024-25Updated 6/8/2026
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Take any composite number. Break it into primes. No matter which pair of factors you start with, you always end up with the same collection of primes. That is the whole theorem.

Statement

Fundamental Theorem of Arithmetic
Every composite number can be written as a product of primes, and this factorisation is unique except for the order in which the prime factors occur.

Prime factorise 156

  1. 1156 = 2 × 78
  2. 278 = 2 × 39
  3. 339 = 3 × 13
  4. 4So 156 = 2² × 3 × 13

Using it for HCF and LCM

Once both numbers are in prime form, HCF takes the smallest power of each shared prime, and LCM takes the largest power of every prime that appears.

HCF and LCM of 96 and 404

  1. 196 = 2⁵ × 3
  2. 2404 = 2² × 101
  3. 3HCF = 2² = 4
  4. 4LCM = (96 × 404) ÷ 4 = 9696
Use this to get the LCM once you know the HCF.

Quick recap

  • Prime factorisation of a composite number is unique
  • HCF: lowest powers of common primes
  • LCM: highest powers of all primes
  • HCF × LCM = product, for two numbers only

NCERT Mathematics, Textbook for Class X, Chapter 1, Section 1.2

Sources