HCF and LCM: buses and bamboo poles
Find when two buses leave the bus park together again (LCM), and how to cut two poles into the longest equal pieces (HCF).
Core concept
In short: Find when two buses leave together again, and the biggest equal groups you can make.
The LCM is the first common multiple and the HCF is the biggest common factor; prime factorisation finds both
Topic: Factors and multiples
At the bus park, one microbus leaves every 4 minutes and another every 6 minutes. They both leave at 0 minutes; when do they leave together again? The times of the first bus are multiple: What you get when you multiply a number by 1, 2, 3 … 12 is a multiple of 3. All key words of 4 (4, 8, 12, …) and of the second, multiples of 6 (6, 12, …). The first time in both lists is the lowest common multiple, the LCM: 12 minutes. A carpenter wants to cut a 12 m and an 18 m bamboo pole into pieces that are all the same length, as long as possible, with nothing wasted. The piece length must divide both lengths: it is the highest common factor: A number that divides another exactly. 3 is a factor of 12. All key words, the HCF.
What you will learn
- Find the LCM of two numbers from their multiples and from prime factors.
- Find the HCF of two numbers from their factors and from prime factors.
- Decide whether a word problem needs the HCF or the LCM.
Before you start
- Multiplication tables and division without remainder.
Materials
- A bus timetable, a ribbon or a stick to cut
Equipment
- A ruler
Procedure
- Tap − or + to change a and b; tap the chip to switch between the buses and the poles.
- With buses, find the first time both leave together; with poles, see the longest equal pieces.
- Record two buses that first leave together after 12 minutes, and two buses that only meet after a × b minutes.
- Cut 12 m and 18 m poles into the longest equal pieces, and find two poles that can only be cut into 1 m pieces.
How this simulation is simplified
- Both buses leave at 0 minutes; pieces are whole metres.