What is the Fundamental Theorem of Arithmetic? It sounds important but I don't really get what it means for solving problems.

The Fundamental Theorem of Arithmetic states that every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur. For example, the number $60$ can be uniquely factorized as $2 \times 2 \times 3 \times 5$, or $2^2 \times 3 \times 5$. You won't find any other set of prime numbers that multiply to $60$. Its importance in problem-solving is immense. It's the theoretical basis for finding the HCF and LCM of numbers using the prime factorization method. It's also crucial for proving the irrationality of numbers like $\sqrt{2}$, as it ensures that if a prime number divides $p^2$, it must also divide $p$. It essentially guarantees that every number has a unique 'building block' structure using prime numbers.