LCM Calculator · Practical guide
LCM Word Problems: Repeating Events
When two cycles repeat at different intervals, the LCM tells you when they line up again.
At a glance
The Key Insight
Any word problem that asks "when will these happen at the same time again?" is an LCM problem. If one event repeats every A units and another every B units, they coincide every lcm(A, B) units. Find the LCM of the intervals and you have your answer.
Interactive tool
LCM Calculator
Enter your own values, then compare the result with the examples below.
Results
Example 1: Two Buses
Bus A leaves the station every 12 minutes and Bus B every 18 minutes. If they depart together at 8:00 AM, when do they next leave together? Compute lcm(12, 18) = 36 minutes — so 8:36 AM, then 9:12 AM, and so on.
Example 2: Hot Dogs and Buns
Hot dogs come in packs of 10 and buns in packs of 8. How many packs of each make the counts match? lcm(10, 8) = 40, so buy 4 packs of hot dogs (40 dogs) and 5 packs of buns (40 buns) — nothing left over.
Example 3: Three Blinking Lights
Three lights blink every 4, 6, and 8 seconds. They flash together at lcm(4, 6, 8) = 24 seconds. For three or more intervals, chain the LCM in pairs — or just type "4, 6, 8" into the calculator above.
How to Recognize an LCM Problem
- Repeating cycles: "every 12 minutes," "every 18 days" — find when they sync.
- Packaging mismatches: items sold in different pack sizes — find the smallest matching count.
- Tiling or cutting: the smallest square that can be tiled by rectangles of a given size.
The tell-tale words are "smallest," "first time together," and "again." If the problem asks for the largest shared piece instead, that is a GCF problem — the mirror image of this one.
Frequently Asked Questions
How do I know if a word problem needs LCM or GCF?
Ask whether you are building up to a common point (LCM) or breaking down into equal parts (GCF). "When do they coincide again" means LCM; "what is the largest equal group or piece" means GCF.
Two events repeat every 6 and 9 days. When do they coincide?
Every lcm(6, 9) = 18 days. The 6-day event happens on days 6, 12, 18 and the 9-day event on days 9, 18 — day 18 is the first match.
Can LCM word problems have three or more cycles?
Yes. Find the LCM of all the intervals by chaining pairs: lcm(4, 6) = 12, then lcm(12, 8) = 24. The calculator accepts the whole list at once.
What if the intervals share no common factor?
Then the LCM is simply their product. Events repeating every 5 and 7 days coincide every 35 days.