When should I use prime factorization to find HCF/LCM, and when should I use Euclid's Division Algorithm? Is one better?

Euclid's Division Algorithm is specifically designed for finding the HCF of two (or more) numbers, especially larger ones, very efficiently. It's a step-by-step process of repeated division. Prime factorization is more versatile; you can use it to find both HCF and LCM. For HCF using prime factorization, you take the product of common prime factors with the lowest power. For LCM, you take the product of all prime factors (common and uncommon) with the highest power. Neither method is inherently 'better'; their utility depends on the specific problem and the size of the numbers. For HCF of very large numbers, Euclid's Algorithm is often quicker. For LCM, prime factorization is usually the preferred method as Euclid's Algorithm doesn't directly yield LCM.