What's the difference between Euclid's Division Lemma and Algorithm? And why do we even learn it?

Euclid's Division Lemma is a proven statement used to prove other statements. It states that for any two positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that $a = bq + r$, where $0 \le r < b$. The Euclid's Division Algorithm, on the other hand, is a series of well-defined steps that uses the lemma repeatedly to find the Highest Common Factor (HCF) of two positive integers. We learn it because it provides a systematic and efficient method to find the HCF of large numbers, which has practical applications in fields like cryptography and computer science, though in Class 10, we primarily focus on its use for HCF calculations and understanding fundamental number theory concepts.