# Prime Factorization

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### What is Prime Factorisation?

Prime factorisation allows us to write any number as a product of prime factors. For example, consider the number 360. Let us write this number as follows:

$\begin{array}{l}360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5\\\;\;\;\;\;\, = {2^3} \times {3^2} \times {5^1}\end{array}$

Basically, we have expressed 360 as a product of prime numbers. Let us take some other (composite) numbers and express them as products of prime numbers:

$\begin{array}{l}\;\,80 = 2 \times 2 \times 2 \times 2 \times 5\\\;\;\;\;\;\,\,\, = {2^4} \times {5^1}\\144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3\\\;\;\;\;\;\,\,\, = {2^4} \times {3^2}\\600 = 2 \times 2 \times 2 \times 3 \times 3 \times 5\\\;\;\;\;\;\,\,\,= {2^3} \times {3^2} \times {5^1}\end{array}$

Expressing a number this way (as a product of primes) is called the prime factorization of that number. Note that the prime factorization of a prime number is trivial, since the only non-unity divisor of any prime number is that number itself.

### Can every number be prime-factorised in a unique way?

Two questions now arise:

1. Can every composite number be expressed as a product of primes, that is, can every composite number be prime factorized?

2. If we prime factorize a composite number, is that factorization unique? That is, if we express a composite number as a product of primes, can it be expressed as a product of some other set of primes?

These questions are answered by one of the most important results in Mathematics, which we discuss next.

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