Maths

HCFs and LCMs

Nici Hoyle

— 2 min read

Highest Common Factors (HCF)

and

Lowest Common Multiples (LCM)

Highest Common Factor example

Tom is organising chairs for a wedding reception.


He has 48 folding chairs and 60 plastic chairs.


He wants to set up rows with both types of chairs in each row, with no leftover chairs.


Each row has to have the same total number of chairs.

What is the maximum number of chairs he can put in each row?

Solution:

To find the maximum number of chairs Tom can put in each row, he needs to find the HCF of 48 and 60.

1. Prime factorisation of 48: 48 = 24 × 3

2. Prime factorisation of 60: 60 = 22 × 3 × 5

To find the HCF, take the greatest number that divides both original numbers: HCF = 22 × 3 = 12.

Answer: Tom can place 12 chairs of each type in each row, and there will be no leftover chairs.

Why This is Useful:

The concept of HCF is incredibly practical when you need to divide items evenly. This could apply to situations like:

  • Food distribution: If you need to distribute food (e.g., pizza slices or portions of a meal) evenly across a group.
  • Packaging: When you're packaging products into boxes and need to ensure each box contains the same number of items.
  • Event planning: For setting up chairs, tables, or decorations where equal groupings are required.

Lowest Common Multiple example

You're organising two regular events at a community centre.


One event happens every 6 days, and the other happens every 8 days.


Both events are scheduled to occur on the same day this coming Sunday.


How many days will it be until both events happen on the same day again?

Solution:

To determine when both events will coincide again, you need to find the LCM of 6 and 8.

1. Prime factorization of 6: 6 = 2 × 3

2. Prime factorization of 8: 8 = 23

To find the LCM, take the highest powers of all prime factors present: LCM = 23 × 3 = 24

Answer: Both events will happen on the same day again in 24 days.

Why This Is Useful:

The concept of LCM is vital in situations where you’re coordinating schedules or timing, such as:

  • Event planning: When organizing recurring activities and you need to find when they overlap.
  • Transportation scheduling: Synchronizing bus, train, or flight schedules.
  • Manufacturing: Coordinating machines or processes with different cycle times.

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