Lowest common multiple

Key Notes:

What is the Lowest Common Multiple?

Lowest Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. It is the first number that both of the given numbers divide into without leaving a remainder.

Key Concepts

Multiple:
  • Definition: A number that can be divided by another number exactly.
  • Example: Multiples of 4 are 4, 8, 12, 16, 20, etc.
Common Multiple:
  • Definition: A number that is a multiple of each of the given numbers.
  • Example: Common multiples of 4 and 6 are 12, 24, 36, etc.
Lowest Common Multiple:
  • Definition: The smallest number that is a multiple of all given numbers.
  • Example: For 4 and 6, the LCM is 12.

How to Find the LCM

There are different methods to find the LCM:

Listing Multiples Method

List the Multiples of Each Number:

Find several multiples for each number.

Example: For 4 and 6:

  • Multiples of 4: 4, 8, 12, 16, 20, 24, …
  • Multiples of 6: 6, 12, 18, 24, 30, 36, …

Find the Smallest Common Multiple:

  • Identify the smallest number that appears in both lists.
  • LCM: 12

Prime Factorisation Method

Prime Factorise Each Number:

Break each number into prime factors.

Example:

  • 4 = 2²
  • 6 = 2 × 3

Take the Highest Power of Each Prime:

  • Find the highest power of all prime factors involved.
  • LCM: 2² × 3 = 12

Using the HCF to Find LCM

Find the HCF of the Numbers:

  • Use the HCF method (Listing Factors or Prime Factorisation).

Use the Formula:

LCM={Product of the Numbers}​ / HCF

Example: For 4 and 6:

  • Product of Numbers: 4×6=24
  • HCF of 4 and 6: 2
  • LCM: 24 /2 =12

Examples

Find the LCM of 8 and 12:

Listing Multiples:

Multiples of 8: 8, 16, 24, 32, 40, 48, …

Multiples of 12: 12, 24, 36, 48, 60, …

LCM: 24

Prime Factorisation:

  • 8 = 2³
  • 12 = 2² × 3
  • Highest Powers: 2³ × 3 = 24

Find the LCM of 9 and 15:

Listing Multiples:

Multiples of 9: 9, 18, 27, 36, 45, …

Multiples of 15: 15, 30, 45, 60, 75, …

LCM: 45

Prime Factorisation:

  • 9 = 3²
  • 15 = 3 × 5
  • Highest Powers: 3² × 5 = 45
Why LCM is Useful

Scheduling Events:

  • Helps in finding when two events will occur together again.

Solving Problems:

  • Useful in problems related to repeating patterns or syncing events.

Adding or Subtracting Fractions:

  • Helps in finding a common denominator.

Practice Problems

  • Find the LCM of 5 and 7.
  • Find the LCM of 10 and 15.
  • Find the LCM of 14 and 21.

Visual Representation

Example: LCM of 4 and 6

Listing Multiples:

Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, ...

Common Multiples:

12, 24, 36, ...
Lowest Common Multiple: 12

Prime Factorisation:

4 = 2²
6 = 2 × 3
LCM = 2² × 3 = 12
Additional Notes
  • LCM vs. HCF: LCM is the smallest number that can be divided by both numbers, while HCF is the largest number that can divide both numbers.
  • LCM and Multiples: LCM is always a multiple of each of the numbers you’re finding it for.

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