The fundamental theorem of arithmetic (also called the unique factorization theorem or the unique prime factorization theorem) states that:
"Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur".
Let us consider the prime factorization of 168 [see adjacent table].
We observe that, in all the three prime factorizations of 168, the prime numbers appearing are same, although the order in which they appear are different.
Thus, the prime factorization of 168 is unique except for order.
In general, given a composite number 'N', we factorize it as:
N = p1 × p2 × . . . × pn
where p1, p2, . . ., pn are primes and written in ascending order
i.e., p1 ≤ p2 . . . ≤ pn
If we combine the same primes, we will get powers of primes.
For example, 168 = 2 × 2 × 2 × 3 × 7 = 23 × 3 × 7