HCF and LCM: Finding the Best Common Ground

Imagine you have 12 stickers and your friend has 18 stickers. You both want to split your stickers into equal groups of the same size — as large as possible. What is the biggest group size that works for both of you? Hold that thought. Now think about two buses: Bus A leaves every 4 minutes, Bus B leaves every 6 minutes. They both just left at the same time. When is the next time they will leave together? Both puzzles have something in common — they use factors and multiples. Today you will learn the exact tools to solve them quickly and reliably.

**Factors and Multiples — a quick reminder** A factor of a number divides into it exactly, with no remainder. A multiple of a number is what you get when you multiply it by a whole number (1, 2, 3, …). **What is the HCF?** HCF stands for Highest Common Factor. It is the largest factor that two (or more) numbers share. You write it as HCF(a, b). To find HCF(12, 18) by listing: • Factors of 12: 1, 2, 3, 4, 6, 12 • Factors of 18: 1, 2, 3, 6, 9, 18 • Common factors (appear in both lists): 1, 2, 3, 6 • The highest of these is 6, so HCF(12, 18) = 6. This answers the sticker puzzle: the biggest equal group size is 6. **What is the LCM?** LCM stands for Lowest Common Multiple. It is the smallest multiple that two (or more) numbers share. You write it as LCM(a, b). To find LCM(4, 6) by listing: • Multiples of 4: 4, 8, 12, 16, 20, … • Multiples of 6: 6, 12, 18, 24, … • The first multiple that appears in both lists is 12, so LCM(4, 6) = 12. This answers the bus puzzle: the two buses next leave together after 12 minutes. **A useful shortcut** For any two numbers a and b: HCF(a, b) × LCM(a, b) = a × b. Check: HCF(12, 18) × LCM(12, 18) = 6 × 36 = 216, and 12 × 18 = 216. ✓ This shortcut lets you find the LCM quickly once you know the HCF: LCM = (a × b) ÷ HCF.