The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic is a very powerful statement that is used in numerous proofs, especially in the realm of number theory. This theorem tells us that there is effectively only one way that we can factorise a whole number into prime numbers being multiplied together, and that any alternatives are just a rearrangement of the order of multiplication.

For every integer $n > 1$ :

There exists a set of distinct prime numbers $p_1, p_2, \dots p_k$ and positive integers $e_1, e_2, \dots e_k$ such that

$n = p_1^{e_1}p_2^{e_2}\dots p_k^{e_k}$

This representation is unique, except for the order of the factors.

For example, the number $60$ can be written as

$60 = 2^2 \cdot 3^1 \cdot 5^1 = 2\cdot2\cdot3\cdot5$

There is no other way to write $60$ as a product of primes that is not just a rearrangement of this factorisation.

...