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.
Practice This ConceptMain explanation
Teacher 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.
Example
84 = 2^2 x 3 x 7 is a prime factorisation of 84.
Simple analogy
Prime only, prime only, until the tree stops.
Common confusion
Students stop at a product of composite factors such as 84 = 6 x 14 and call it prime factorisation.
Exam tip
Always break numbers completely into primes before using them in HCF or LCM questions.
Answer writing and exam use
1-mark use
Write the exact meaning of fundamental theorem of arithmetic in one clean line.
2-mark use
Define fundamental theorem of arithmetic and add one example or condition.
3-mark use
Explain fundamental theorem of arithmetic, show the method or example, and mention the common mistake.
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.
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