Divisibility Proof Examples

Number Theory · Axiom Academy

EXAMPLE Divisibility Proof Examples Master the techniques for proving divisibility properties with five complete worked examples Excellent work! You've completed five fundamental divisibility proofs. Here's what we learned: Divisibility Definition: To prove a | b , we must show that b = ak for some integer k . Linear Combinations: If a divides two numbers, it divides any linear combination of them. Case Analysis: For proofs involving all integers, consider different cases (e.g., remainders modulo 3). Consecutive Integers: Products of consecutive integers have useful divisibility properties due to their structure. Counterexamples: To disprove a universal statement, find just one counterexample. Remember that a | bc does NOT imply a | b or a | c . Proof Strategy: Start with definitions, use substitution, factor expressions, and conclude by showing the desired form. These proof techniques are fundamental to number theory. Practice identifying which technique applies to different divisibility problems!

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