Division Algorithm Examples

Number Theory · Axiom Academy

EXAMPLE Division Algorithm Examples Master the division algorithm through worked examples with positive, negative, and applied scenarios Excellent work! You've completed all five division algorithm examples. Here's what we learned: Division Algorithm Formula: For any integers a and b > 0 , we can write a = bq + r where 0 ≤ r . This always guarantees a unique quotient and remainder. Handling Negative Dividends: When a , we must ensure the remainder r is still non-negative. This often requires adjusting the quotient down and the remainder up. Remainder Constraint: The remainder must always satisfy 0 ≤ r . This is crucial for uniqueness and distinguishes the division algorithm from simple division with negative remainders. Verification Method: Always check your answer by verifying (1) a = bq + r and (2) 0 ≤ r . Both conditions must hold. Modular Arithmetic Connection: The remainder r from the division algorithm is the same as a mod b , which is useful for cyclic problems like days of the week or clock arithmetic. The division algorithm is foundational in number theory. Practice with various values to build confidence, especially with negative dividends!

This is the written version of the interactive lesson above. See the full Number Theory course.