medium difficulty5 min

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.

In one line: Prime only, prime only, until the tree stops.

Practice This Concept

Understand it

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.

Example

84 = 2² × 3 × 7 is a prime factorisation of 84.

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

Method you can reuse

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

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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

Common wrong answer

Stopping at 2 × 30.

Why it loses marks: 30 is still composite and must be split further.

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

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

1
Enough for 1 mark
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
Add for 2 marks
Example: 84 = 2² × 3 × 7 is a prime factorisation of 84.
3
Add for 3 marks
Once a number is broken fully into prime factors, the same prime structure can be compared across numbers without confusion.

MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results