How this instrument works
Picture two meshed gears, one with 12 teeth and one with 18, each carrying a single marked tooth that starts lined up against its partner. As the gears turn, that marked pair drifts apart mesh by mesh and only lines back up once both wheels have separately completed a whole number of full rotations at the same instant — never partway through a turn for either one. The number of meshes that takes is the least common multiple of the two tooth counts, the smallest number both counts divide into evenly.
The arithmetic is identical to this site's plain LCM calculator, which takes two abstract whole numbers and returns their smallest shared multiple with no picture attached. This page anchors that same calculation to one specific mechanical scenario: two meshed wheels with different tooth counts, and the question of exactly when their marked teeth meet again. Naming the page 'least common factor' rather than 'least common multiple' matches how some people phrase the search once they already know roughly what answer they're after, even though the counts being combined build toward a shared multiple, not a shared divisor.
Wheels whose tooth counts share no common factor beyond 1 take the longest possible path back to realignment: their marked teeth meet again only after a number of meshes equal to the straight product of both counts, since nothing smaller is divisible by both. Two identical gears sit at the opposite extreme — with matching tooth counts, the marked pair lines back up at the very next full rotation, every single time.
- Enter the first gear's tooth count into the First gear's tooth count field.
- Enter the second gear's tooth count into the Second gear's tooth count field.
- Read Meshes until both realign: how many tooth-meshes pass before the two marked teeth meet again.
- Divide that result by each tooth count separately to see how many full rotations each individual wheel completes by then.
Worked example — 12-tooth and 18-tooth gears
A 12-tooth gear meshes with an 18-tooth gear, marked tooth to marked tooth at the start. Their least common multiple is 36: 36 ÷ 12 = 3, so the smaller wheel completes exactly 3 full rotations by then, and 36 ÷ 18 = 2, so the larger wheel completes exactly 2 full rotations — both whole numbers of turns finishing at precisely the same mesh, the 36th, which is where the marked teeth meet again.
Gears with 5 and 7 teeth share no common factor at all, so they realign only after their straight product, 35 meshes — the longest wait two counts that size can produce. Two matching 6-tooth gears sit at the other end: their least common multiple is just 6, so the marked pair lines back up after a single full rotation each.
Questions
What does the least common factor of two gears mean?
It is the number of tooth-meshes that pass before two meshed gears' marked teeth line up again, found as the least common multiple of their two tooth counts. Gears with 12 and 18 teeth realign every 36 meshes.
Why is this page called least common factor instead of least common multiple?
Because that is how some people phrase the search when they already know roughly what they're after, even though the calculation underneath is a least common multiple — the smallest shared multiple of the two tooth counts, not a shared divisor. The math this page runs is identical to a least-common-multiple calculation either way.
How is this different from this site's plain LCM calculator?
Nothing changes in the underlying formula. This page frames the same least-common-multiple calculation around one specific mechanical picture — two meshed gears with different tooth counts realigning — rather than presenting two bare numbers with no attached scenario.
What happens when two gears share no common tooth-count factor?
They take the longest possible time to realign: a number of meshes equal to the straight product of both tooth counts. Gears with 5 and 7 teeth, sharing nothing, realign only after 35 meshes, since nothing smaller than their product is divisible by both.
What if both gears have the same number of teeth?
They realign after every single rotation. Two matching 6-tooth gears have a least common multiple of exactly 6, meaning the marked pair meets again as soon as both wheels complete one full turn.
How many rotations does each gear complete before realigning?
Divide the meshes-until-realign result by each gear's own tooth count. For 12 and 18 teeth realigning at 36 meshes, the smaller wheel turns 36 ÷ 12 = 3 full rotations and the larger turns 36 ÷ 18 = 2, finishing together at the same mesh.