HCF × LCM = product of the numbers
Pair the beads and the HCF and LCM together hold exactly the beads of the two numbers.
A surprising multiplication
4 = 2 × 2 and 6 = 2 × 3. Their HCF is 2 and their LCM is 12.
Multiply the HCF by the LCM: 2 × 12. Which number do you get?
What this lesson covers
The idea
For two numbers, HCF × LCM = Product of the two numbers.
A surprising multiplication
4 = 2 × 2 and 6 = 2 × 3. Their HCF is 2 and their LCM is 12.
Multiply the HCF by the LCM: 2 × 12. Which number do you get?
- 24, which is 4 × 6
- 12
- 8
Pair the beads, then compare
Pair every bead of the top number with an equal bead below. The pairs make the HCF, and every bead once makes the LCM. Then line up HCF × LCM beside the two numbers.
The same beads, counted again
105 × 95 = 3 × 5 × 5 × 7 × 19. The LCM is 3 × 5 × 7 × 19 = 1995. So 105 × 95 = LCM × 5, and 5 is the HCF. The table shows that HCF × LCM and the product of the two numbers give the same answer:
Why? A bead shared by both numbers is in the HCF and in the LCM, which is twice, and it is in both numbers, which is also twice. A bead in only one number is once in the LCM and once in the numbers. The counts match.
For two numbers, HCF × LCM = Product of the two numbers.
- Numbers | HCF × LCM | Both give
- 105, 95 | 5 × 1995 | 9975
- 45, 105 | 15 × 315 | 4725
- 275, 352 | 11 × 8800 | 96800
Notes
For two numbers, HCF × LCM = product of the two numbers: the beads of the HCF and the LCM together are exactly the beads of the two numbers.
Check yourself
Two numbers have HCF 12 and LCM 180. What is the product of the two numbers?
Answer: 2160
12 × 180 = 2160. (The numbers are 36 and 60: 36 × 60 = 2160.)
The product of two numbers is 2400 and their HCF is 20. What is their LCM?
Answer: 120
20 × LCM = 2400, so LCM = 2400 ÷ 20 = 120. (The numbers are 40 and 60.)
8 = 2 × 2 × 2 and 15 = 3 × 5 share no prime. What is their LCM?
Does HCF × LCM = product also work for three numbers? Test 4, 6, 8: HCF 2, LCM 24, product 192.
- 120, the product 8 × 15 — correct. Yes! The HCF is 1, so 1 × LCM = 8 × 15 = 120. With no shared beads, every bead is used once.
- 1. The HCF is 1. The LCM must be a multiple of 8 and of 15.
- 23. 23 is 8 + 15. The LCM is a multiple of both numbers.
- No. 2 × 24 = 48, but 4 × 6 × 8 = 192. — correct. Yes! The rule is for two numbers. One example that fails is enough to show that it does not hold for three.
- Yes, 2 × 24 = 192.. Check: 2 × 24 = 48, not 192.
- Yes, because 2 + 24 = 26.. The rule multiplies the HCF and LCM, and 2 × 24 = 48, which is not 192.