Fundamental Theorem of Arithmetic
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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.