Prime Factorization and Its Consequences

State and prove the Fundamental Theorem of Arithmetic using Euclid's Lemma.

Explain the conceptual significance of unique factorization and contrast with systems where it fails.

Derive the formulas for the number and sum of positive divisors of an integer from its prime factorization.

Characterize perfect squares and other powers by analyzing their prime factorization structure.

Prove the infinitude of primes using Euclid's argument and variations.