TITLE: Why is the Fundamental Theorem of Arithmetic so important?
QUESTION [18 upvotes]: I've recently read about the Fundamental Theorem of Arithmetic and I think that I have just about understood the proof. What I found quite interesting at first was the "Fundamental" part in the name. I understand that my mathematical experience is nowhere near enough to fully appreciate this theorem, but can someone can explain to me why this theorem is so important? What would happen if this theorem wasn't actually true?
Thanks in advance!

REPLY [5 votes]: I decided to compare this theorem to another favorite theorem of mine, Fermat's little theorem (which some people just call "Fermat's theorem).
On ProofWiki, I went to the pages for both theorems and clicked "what links here": https://proofwiki.org/wiki/Special:WhatLinksHere/Fundamental_Theorem_of_Arithmetic , https://proofwiki.org/wiki/Special:WhatLinksHere/Fermat%27s_Little_Theorem
As you can see, the fundamental theorem of arithmetic is invoked for the following results:

Positive Integer Greater than 1 has Prime Divisor
Integers form Unique Factorization Domain
Cartesian Product of Countable Sets is Countable
Power of Prime Divides
Set of Divisors of Integer
Tau Function from Prime Decomposition
Ostrowski's Theorem
Sum of Reciprocals of Primes is Divergent
Zero and One are the only Consecutive Perfect Squares
Product Form of Sum on Completely Multiplicative Function
Solution of Linear Diophantine Equation
Number of Primes is Infinite (this can be proven without the fundamental theorem of arithmetic, so it's okay if you choose to cross it off the list)
Dirichlet L-Function from Trivial Character
Finite Multiplicative Subgroup of Field is Cyclic
Sum Over Divisors of von Mangoldt is Logarithm
Power of Elements is Subgroup
Power of Sum Mod Prime
Set of Finite Subsets of Countable Set is Countable
Cyclicity Condition for Units of Ring of Integers Modulo m
Irrationality of Logarithm
Integer to Rational Power is Irrational iff not Integer or Reciprocal

Compare Fermat's little theorem:

Floor Defines Equivalence
Index of Subgroup of Subgroup
Ring of Integers Modulo Prime is Field
Wilson's Theorem
Lucas-Lehmer Test
Integer as Sum of Two Squares
Factors of Mersenne Numbers
Values of Legendre Symbol
P-adic Norm not Complete on Rational Numbers
Euler's Criterion
Finite Cyclic Group has Euler Phi Generators

I have chosen to ignore lemmas, corollaries, definitions, biographies, and of course pages that are very specific to ProofWiki, like the Help:Page Editing page.
My conclusion is, that by this measure, Fermat's little theorem is important in the sense that other theorems depend on it, but not as important as the fundamental theorem of arithmetic.