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Fundamental Theorem of Arithmetic

Every composite number can be written as a product of prime numbers, and this prime factorisation is unique apart from the order of the primes.

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Student-friendly explanation

This theorem is the backbone of HCF, LCM, and decimal expansion work in Class 10. Once a number is broken fully into prime factors, the same prime structure can be compared across numbers without confusion. The order may change, but the prime factors themselves do not.

How to write this in exams

  1. 1

    Start with the exact idea

    Every composite number can be written as a product of prime numbers, and this prime factorisation is unique apart from the order of the primes.

  2. 2

    Then show how to use it

    1. Start with the composite number. 2. Split it into two factors. 3. Keep splitting until every factor is prime. 4. Rewrite repeated primes using powers. 5. Check that no composite factor remains.

  3. 3

    Add one concrete example

    84 = 2^2 x 3 x 7 is a prime factorisation of 84.

  4. 4

    Avoid this incomplete answer

    Stopping at 2 x 30 is wrong because 30 is still composite and must be split further.

Definition

Every composite number can be written as a product of prime numbers, and this prime factorisation is unique apart from the order of the primes.

Example

84 = 2^2 x 3 x 7 is a prime factorisation of 84.

Rule to remember

Every composite number has a unique prime factorisation apart from the order of factors.

Memory hook

Prime only, prime only, until the tree stops.

Examples and method

Worked example

60 = 2 x 30 = 2 x 2 x 15 = 2 x 2 x 3 x 5 = 2^2 x 3 x 5.

Method to apply

1. Start with the composite number. 2. Split it into two factors. 3. Keep splitting until every factor is prime. 4. Rewrite repeated primes using powers. 5. Check that no composite factor remains.

Diagram support

A factor tree is the easiest way to show how a number breaks into primes.

How CBSE asks it

State the theorem, use a factor tree, or use prime factorisation while finding HCF and LCM.

Avoid common mistakes

Common confusion

Students stop at a product of composite factors such as 84 = 6 x 14 and call it prime factorisation.

Common wrong answer

Stopping at 2 x 30 is wrong because 30 is still composite and must be split further.

Exam tip

Always break numbers completely into primes before using them in HCF or LCM questions.

Quick check

Why is 72 = 8 x 9 not the final form for prime factorisation?

Because 8 and 9 are not prime numbers, so the number must be broken further until only primes remain.

Answer writing and exam use

1-mark answer

Every composite number can be written as a product of prime numbers, and this prime factorisation is unique apart from the order of the primes.

2-mark answer

Every composite number can be written as a product of prime numbers, and this prime factorisation is unique apart from the order of the primes. Every composite number has a unique prime factorisation apart from the order of factors. 84 = 2^2 x 3 x 7 is a prime factorisation of 84.

3-mark answer

This theorem is the backbone of HCF, LCM, and decimal expansion work in Class 10. Once a number is broken fully into prime factors, the same prime structure can be compared across numbers without confusion. The order may change, but the prime factors themselves do not. Every composite number has a unique prime factorisation apart from the order of factors. 60 = 2 x 30 = 2 x 2 x 15 = 2 x 2 x 3 x 5 = 2^2 x 3 x 5. State the theorem, use a factor tree, or use prime factorisation while finding HCF and LCM. Stopping at 2 x 30 is wrong because 30 is still composite and must be split further.
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