No bead is lost

The idea

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b, so the LCM can be found from the HCF; for three numbers the product is not equal to HCF × LCM.

Multiply HCF by LCM

For 6 and 20, the HCF is 2 and the LCM is 60. And 6 × 20 = 120.

What do you get when you multiply HCF × LCM?

• 120, the same as 6 × 20
• 62
• Something else

Regroup the beads

Tap a row to regroup: the shorter lane goes to the HCF, the longer lane to the LCM. Count the beads before and after. The last pair has three numbers.

Same beads, same product

For any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b.

Why? In every prime row the two lanes hold the same beads as the HCF lane and the LCM lane together, so nothing is lost. This gives a shortcut: LCM = (a × b) ÷ HCF.

Example: HCF(96, 404) = 4, so LCM = 96 × 404 ÷ 4 = 9696.

With three numbers the middle lane is left behind. So 6 × 72 × 120 is not HCF × LCM.

Notes

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. So LCM = (a × b) ÷ HCF. For three numbers, the product is not equal to HCF × LCM.

Check yourself

HCF(12, 18) = 6. Use it to find LCM(12, 18).

Answer: 36

12 × 18 = 216 and 216 ÷ 6 = 36.

HCF(25, 40) = 5. Find LCM(25, 40).

Answer: 200

25 × 40 = 1000 and 1000 ÷ 5 = 200.

Two numbers have HCF 9 and LCM 360. One of them is 45. What is the other?

Answer: 72

9 × 360 = 3240 and 3240 ÷ 45 = 72.

Which statement is true?

• For any three positive integers, HCF × LCM = their product. Not true. For 6, 72 and 120, 6 × 72 × 120 = 51840 but HCF × LCM = 6 × 360 = 2160.
• For any two positive integers a and b, HCF × LCM = a × b — correct. Yes! Every bead of a and b ends up either in the HCF or in the LCM.
• HCF × LCM is always smaller than a × b. It is equal to a × b for two numbers.
• HCF × LCM = a + b. Multiplication, not addition. Check with 6 and 20: 2 × 60 = 120.