The Division Algorithm is a fundamental principle in mathematics that states for any integers a and b (where b is not zero), there exist unique integers q (the quotient) and r (the remainder) such that a = b × q + r. The remainder r must be non-negative and less than the absolute value of b. This algorithm is essential for various mathematical operations, including simplifying fractions and performing long division. It also plays a crucial role in number theory, particularly in concepts like greatest common divisor and modular arithmetic.