number theoryleast common multiplemodular arithmetic
Problem Statement
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise.
In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?<spanclass=′latex−bold′>(A)</span>6<spanclass=′latex−bold′>(B)</span>8<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>14<spanclass=′latex−bold′>(E)</span>24