Problem
COM-B2-M05-P004 Steps on a Line
#4
★★☆☆☆ Level 2 of 5
A token starts at point \(0\). In one move it may be shifted \(10\) units to the right or \(15\) units to the left. Prove that it will never reach point \(2026\).
Both moves are multiples of \(5\).
Each move changes the coordinate by a multiple of \(5\). Therefore the coordinate modulo \(5\) is invariant.
The initial coordinate has residue \(0\), while \(2026\equiv 1\pmod 5\). Hence point \(2026\) is unreachable.
The problem teaches students to see the gcd of step sizes as a modular invariant.