LCM & HCF

HCF is the largest number that divides them all. LCM is the smallest number they all divide.

The whole topic turns on picking the right one, so learn the signal words:

• "again together", "next time", "least number of…" → LCM (bells tolling, lights blinking, runners meeting).
• "greatest measure", "largest tile", "maximum number of equal groups" → HCF.

Three rules do the arithmetic:

1. HCF × LCM = product of the two numbers — the fastest way to get the second number when the first is known.
2. LCM(a, b) = a × b ÷ HCF(a, b).
3. Least number leaving the same remainder r with a, b, c → LCM(a, b, c) + r.

The commonest mistake is answering an "again together" question with the HCF. A quick guard: the LCM is never smaller than the largest number, and the HCF is never larger than the smallest. If your answer breaks that, you picked the wrong one.

HCF × LCM = a × b · LCM = a×b/HCF · Same remainder r → LCM + r · "together again" = LCM, "greatest measure" = HCF

Three bells toll at intervals of 9, 12 and 15 seconds. If they toll together now, when next? → "Together again" means LCM. 9 = 3², 12 = 2²×3, 15 = 3×5, so LCM = 2²×3²×5 = 180 seconds (3 minutes). Sanity check: 180 ÷ 9 = 20, 180 ÷ 12 = 15, 180 ÷ 15 = 12 — all whole numbers, so all three really do toll at 180 s.