Euclid's division lemma for integers a, b > 0 says there exist q, r such that:

Euclid's Division Lemma is a fundamental principle in number theory that formalizes the process of division. ### Formal Statement For any two positive integers $a$ (the dividend) and $b$ (the divisor), there exist unique integers $q$ (the quotient) and $r$ (the remainder) such that: $$a = bq + r$$ ### The Remainder Constraint The remainder $r$ must satisfy a specific condition to ensure uniqueness: 1. It must be non-negative: $r \geq 0$. 2. It must be strictly less than the divisor: $r < b$. Combining these, we get the constraint: $0 \leq r < b$. ### Conclusion Comparing this to the given options: - **$a = bq + r, 0 \leq r < b$** is the standard mathematical definition of the lemma. - The other options represent incorrect algebraic relationships or miss the critical remainder constraint.