Finding Common Ground · Lesson 6 of 7
Efficient Joint Methods and the Product Relationship
“Calculate HCF and LCM together and explain their two-number product link.”
• Use shared division to find both HCF and LCM. • Understand why a larger common composite divisor is allowed. • Explain why LCM cannot exceed the product of two positive numbers. • Use and justify HCF × LCM = product for two numbers, and identify its three-number limit.
One division layout, two results
Place two numbers side by side and repeatedly divide both by the same common prime. Stop when the remaining numbers have no common factor greater than 1. The product of the shared divisors is their HCF. To make a number divisible by both originals, include the final two coprime remainders as well; that gives their LCM.
Problem
Find the HCF and LCM of 84 and 180 by shared division.
- 1.Divide both by 2: 42, 90; again by 2: 21, 45; then by 3: 7, 15.
- 2.The remaining 7 and 15 are coprime. HCF = 2 × 2 × 3 = 12.
- 3.LCM = 12 × 7 × 15 = 1260; both original numbers divide it.
You can divide by a common composite factor when you spot one. For 300 and 150, divide both by 50 to get 6 and 3; divide both again by 3 to get 2 and 1. The left-side product 50 × 3 is 150, and including 2 × 1 gives LCM 300. A common factor need not be prime if the reasoning and final coprime check are sound.
Problem
Find the HCF and LCM of 630 and 770.
- 1.Both divide by 10, leaving 63 and 77; both divide by 7, leaving 9 and 11.
- 2.Since 9 and 11 are coprime, HCF = 10 × 7 = 70.
- 3.LCM = 70 × 9 × 11 = 6930.
A relationship between two positive numbers
The product a × b is a common multiple of a and b, so their LCM cannot be greater than that product. The product contains two copies of every prime piece shared by a and b. The LCM uses the larger number of copies; multiplying by the HCF supplies the remaining shared copies. Thus each prime occurs just as often as it does in a × b.
Problem
Find the LCM of 105 and 95 after finding their HCF.
- 1.105 = 3 × 5 × 7 and 95 = 5 × 19, so HCF = 5.
- 2.The product is 105 × 95 = 9975. Divide by the HCF: 9975 ÷ 5 = 1995.
- 3.LCM = 1995 = 3 × 5 × 7 × 19; verify both divisions.
Testing three numbers is a useful investigation, not a licence to reuse a two-number formula. For 2, 3 and 4, the HCF is 1 and LCM is 12, while the product is 24. This counterexample shows the unchanged identity fails.
If the final two numbers still share a factor, the left-side product is not yet the HCF. Continue or explicitly include their remaining HCF.
Quiz
In shared division, what gives the HCF after coprime remainders are reached?
For 84 and 180, the HCF is
For 84 and 180, the LCM is
If HCF of a and b is 5 and a × b = 500, their LCM is
Does HCF × LCM = product hold unchanged for every group of three numbers?
Practice Problems
- Use shared division to find the HCF and LCM of 90 and 150.
- Use a common composite factor to find both values for 84 and 132.
- Explain why the LCM of two positive numbers can never exceed their product.
- For 45 and 105, find both values and verify the product relationship.
- For 275 and 352, find HCF and LCM and verify the relationship.
- Test the two-number identity on 2, 3 and 4 and explain why it fails unchanged.
Key Takeaways
• Shared division by common factors reveals HCF in the left-side product. • Multiply that product by both final coprime remainders to obtain LCM. • A convenient common composite divisor works as well as a prime divisor. • The product of two numbers is a common multiple, so LCM cannot exceed it. • For two positive integers, HCF × LCM equals their product; this statement is not generally true for three.