Fundamental Theorem of Arithmetic: Real Numbers Chapter 1 Class 10 Mathematics CBSE (NCERT)

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.

The Fundamental Theorem of Arithmetic says that every composite number can be factorised as a product of primes. Actually, it says more. It states that given any composite number, it can be factorised as a product of prime numbers in a ‘unique’ way, except for the order in which the primes occur. That is, given any composite number, there is one and only one way to write it as a product of primes, as long as we are not particular about the order in which the primes occur. So, for example, we regard:

2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other possible order in which these primes are written.

This fact is also stated in the following form: The prime factorisation of a natural number is unique, except for the order of its factors.

In general, given a composite number x, we factorise it as:

x = p₁ × p₂ × ... × p_n

where p₁, p₂, ..., p_n are primes and written in ascending order, i.e.,

p₁ ≤ p₂ ≤ ... ≤ p_n

If we combine the same primes, we will get powers of primes. For example:

32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13

or in exponent notation:

32760 = 2³ × 3² × 5 × 7 × 13

Once we have decided that the order will be ascending, then the way the number is factorised is unique.