The Fundamental Theorem of Arithmetic states that every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique apart from the order in which the prime factors occur. This powerful theorem is the basis for computing HCF and LCM.
Every integer greater than 1 either is a prime itself or can be represented as a product of prime numbers, and this representation is unique up to the order of the factors. For example, 1200 = 2⁴ × 3 × 5² — no other set of primes will multiply together to give 1200.
To find the prime factorisation of a number, start dividing by the smallest prime (2). Continue dividing until you cannot divide evenly, then move to the next prime (3, 5, 7, 11…). Write the number as a product of all the prime divisors with their powers.
To find HCF of two or more numbers: (1) Find the prime factorisation of each number. (2) Identify the common prime factors. (3) Take the smallest power of each common prime factor. (4) Multiply these together to get the HCF.
To find LCM of two or more numbers: (1) Find the prime factorisation of each number. (2) List all prime factors that appear in any of the factorisations. (3) Take the greatest power of each prime factor. (4) Multiply these together to get the LCM.
For any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b. This relationship is extremely useful for solving problems where one of the four values is unknown.
By convention, 1 is excluded from the Fundamental Theorem of Arithmetic. If 1 were prime, the uniqueness of prime factorisation would fail since we could multiply by any number of 1s.
Prime factorisation is the basis of modern cryptography (RSA encryption), is used in simplifying fractions, finding common denominators, and solving scheduling and distribution problems.
Use the factor tree method or repeated division by the smallest primes. Start with 2, then 3, 5, 7, and so on. For larger numbers, check divisibility by primes up to the square root of the number.
Book a Trial + Diagnostic session. Get a personalized Learning Path with clear milestones, tutor match, and a plan recommendation — all within 24 hours.
Book Trial + Diagnostic →