Pigeonhole principle — Remainders modulo five

Question

Prove that among any six integers, two have a difference divisible by 5.

This practice question was inspired by Question 14(a) in the 2025 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: identify the objects and boxes, then show why avoiding a repeated box is impossible.The five boxes are remainder classes .
Simplify the previous line carefully, keeping exact values where possible.Place each of six integers into its remainder class.
Simplify the previous line carefully, keeping exact values where possible.Two must share a class.
Finish the calculation, then check that the result meets the question’s conditions.Their difference has remainder zero modulo 5.
State the final answer clearly in the form the question requested.Final answer: Two integers share one of the five possible remainders modulo 5.

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Numbers with equal remainders differ by a multiple of five.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.