Two things that repeat on different cycles come back into step at the lowest common multiple of their periods. That one idea explains a surprising range of practical problems, and the arithmetic behind it is from primary school.
LCM(a, b) = a × b ÷ GCD(a, b)
The GCD and LCM calculator handles any list of numbers. For 24, 36 and 60 it gives a greatest common divisor of 12 and a lowest common multiple of 360.
Gears, and why engineers avoid round numbers
Take an 8 tooth pinion driving a 12 tooth gear. Their LCM is 24, so after 24 tooth engagements, which is 3 turns of the small gear and 2 of the large, the same two teeth meet again. Each tooth on the pinion only ever touches three of the twelve teeth on the gear, and always the same three.
Any manufacturing defect or hard spot therefore lands on that same narrow set of partners every time, and the wear concentrates. Now change the gear to 13 teeth. Eight and thirteen are coprime, so the LCM is 104, and a given tooth pair meets once every 13 turns of the pinion. Every tooth meets every other tooth before anything repeats, and the wear spreads evenly.
That is the hunting tooth principle, and it is why gear tooth counts in a well-designed train are so often prime or at least coprime. The calculator reports whether a set is coprime directly, which is the property that actually matters here rather than the LCM itself.
Calendars and pay cycles
A fortnightly pay cycle is 14 days and a year is 365, which do not divide. Fourteen goes into 365 just over 26 times, so most years have 26 pay days and the remainder accumulates until a year has 27. Employers who budget 26 fortnights as twelve months of salary discover this roughly every eleven years, usually in the week it happens.
Weekly pay is worse, because 52 weeks is 364 days and the extra day accumulates faster, producing a 53 week year about every five or six. The 26 payment year covers what that does to a budget built on monthly figures.
The deeper version of the same problem is the calendar itself, where a solar year is not a whole number of days and the entire leap year system is an attempt to make two cycles line up that genuinely do not. Dates are harder than they look goes through what that costs in practice.
The divisor is the other half of the answer
Where the LCM tells you when cycles coincide, the greatest common divisor tells you the largest piece that fits both without remainder. For a floor of 24 by 36 units, the GCD of 12 says the largest square tile that lays whole in both directions is 12 units, which is a genuinely useful thing to know before buying anything.
The two are tied together by the identity above: for any pair, the GCD multiplied by the LCM equals the product of the numbers. For 8 and 12 that is 4 times 24, which is 96, and 8 times 12 is also 96. The relationship holds for every pair, which is a pleasant thing to be able to check in your head.
Three numbers behave the same way. For 20, 24 and 30 the calculator gives a GCD of 2 and an LCM of 120, so three machines on 20, 24 and 30 minute cycles all start together every two hours, and the shared factor of 2 is the only rhythm they have in common before that.