Unit 3 · Topic 05 · Playing with Numbers
Anaya's bus comes every 4 minutes. Kabir's bus comes every 6 minutes. Both left the stop together at 9:00 am.
Ms. Rao asked: at what time will both buses next be at the stop together?
Listing out arrival times for each bus, one number appeared in both lists first — and that number had a name.
"List every time Anaya's bus arrives after 9:00," said Ms. Rao. Anaya wrote: 4, 8, 12, 16, 20, 24 minutes later. "These are the MULTIPLES of 4 — what you get multiplying 4 by 1, 2, 3, 4..."
Kabir listed his bus's arrivals: 6, 12, 18, 24 minutes later — multiples of 6. "Now find numbers in BOTH lists," said Ms. Rao. Kabir compared: 12 appeared in both, and so did 24. "Those are COMMON multiples of 4 and 6."
"The buses will next meet together at the SMALLEST common multiple," said Ms. Rao. "That's 12 minutes after 9:00 — so 9:12 am." This smallest common multiple is called the Least Common Multiple, or LCM.
Anaya tried another pair: multiples of 3 (3,6,9,12,15,18) and multiples of 5 (5,10,15,20). The smallest number in both lists was 15. "So the LCM of 3 and 5 is 15," she said — useful for scheduling anything that repeats on two different cycles.
Ms. Rao then asked a trickier question: "What if a THIRD friend's bus comes every 8 minutes, also starting at 9:00?" Kabir listed multiples of 8: 8, 16, 24, 32. Comparing all three lists — multiples of 4, of 6, and of 8 — the smallest number appearing in every single one was 24. "So all three buses meet again at 9:24 am," said Kabir, "even though each bus runs on its own separate schedule."
"Notice something useful," said Ms. Rao. "You never had to know exactly HOW MANY buses there would be, or wait around watching a clock. Listing multiples turned a real scheduling puzzle into pure number-hunting." Anaya realised the same idea explains why some festival dates or repeating patterns line up only once every several years — whenever two or more cycles are involved, their next shared moment is always their Least Common Multiple.
A multiple of a number is what you get multiplying it by 1, 2, 3, and so on: 4, 8, 12, 16... are multiples of 4.
A common multiple of two numbers appears in both of their multiple lists.
The Least Common Multiple (LCM) is the SMALLEST common multiple of two or more numbers.
To find the LCM by listing: write out multiples of each number until you spot the smallest one shared by all lists.
List the first six multiples of 5 and of 7. Find their LCM.
Explain why the LCM of two numbers can never be smaller than the larger of the two numbers.
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