> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/501-number-theory/topics/fundamental-theorem-of-arithmetic.md). # Fundamental Theorem of Arithmetic In number theory, **the fundamental theorem of arithmetic**, also called **the unique factorization theorem** or **the unique-prime-factorization theorem**, states that every integer greater than 1 is either a prime number, or it can be represented as the product of prime numbers. Moreover, this representation is unique, up to the order of the factors. Every positive integer n > 1 can be represented in exactly one way as a *product of prime powers*. ∀n ∈ ℕᐩᐩ. n ∈ ℙ ⋁ 🖕 n = ∏ pₙʲ n = p₁ᵃ · pᵇ · pᶜ · … · pₙʲ = ∏ pₙʲ The primes, raised to some power, are the factors of any composite number. It can be said that any composite number is a factor of powers of *all the prime numbers*; the primes not needed for factorizing a particular number can be raised to the zeroth power. ℙ = {2,3,5,7,11,13,...} 2 = 2¹ oo 3 = 3¹ ooo 4 = 2² oo oo 5 = 5¹ ooooo 6 = 2¹ 3¹ ooo ooo 7 = 7¹ ooooooo 8 = 2³ oo oo oo 9 = 3² ooo ooo ooo 10 = 2¹ 5¹ 11 = 12 = 13 = 14 = The **Euclid's lemma**: if a prime `p` divides the product of two integers `ab`, then `p` must divide at least one of them.