Problem
Gear A comes home every 24 teeth and gear B every 18, so the pair meets again at a multiple of both.
A fourth grader can list the multiples of each number and watch for the first one they share.
4.OA.B.4Solve An Easier Related ProblemThe two multiple lists first agree at 72, so 72 teeth bring both gears home at once.
The first number on both 'comes-home' lists is exactly the least common multiple of the two tooth counts.
6.NS.B.4Look For A PatternThe same two teeth meet again once 72 teeth have passed the contact point.
Each gear comes back to its starting tooth only after a whole number of its own teeth have gone by, gear A every 24 and gear B every 18.
The original pair can touch again only at a tooth count that brings both gears home at once, and 72 is the smallest such count.
A count that brings both gears home is built from whole runs of 24 teeth and also from whole runs of 18 teeth, so it is a multiple of each.
Dividing 72 by each gear's own tooth count gives 3 turns for A and 4 for B.
Knowing the shared tooth count, simple division tells how many whole spins each wheel made.
4.OA.C.5Make A Systematic ListMeshed gears line up again at the least common multiple of their tooth counts, then divide it by each gear's teeth to count the turns.