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.

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Key Concepts

1. Multiple:
   • Definition: A number that can be divided by another number exactly.
   • Example: Multiples of 4 are 4, 8, 12, 16, 20, etc.
2. 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.
3. Lowest Common Multiple:
   • Definition: The smallest number that is a multiple of all given numbers.
   • Example: For 4 and 6, the LCM is 12.

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How to Find the LCM

There are different methods to find the LCM:

1. Listing Multiples Method

1. 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, …
2. Find the Smallest Common Multiple:
   • Identify the smallest number that appears in both lists.
   • LCM: 12

2. Prime Factorisation Method

1. Prime Factorise Each Number:
   • Break each number into prime factors.
   • Example:
     • 4 = 2²
     • 6 = 2 × 3
2. Take the Highest Power of Each Prime:
   • Find the highest power of all prime factors involved.
   • LCM: 2² × 3 = 12

3. Using the HCF to Find LCM

1. Find the HCF of the Numbers:
   • Use the HCF method (Listing Factors or Prime Factorisation).
2. 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

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Examples

1. 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
2. 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

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Why LCM is Useful

1. Scheduling Events:
   • Helps in finding when two events will occur together again.
2. Solving Problems:
   • Useful in problems related to repeating patterns or syncing events.
3. Adding or Subtracting Fractions:
   • Helps in finding a common denominator.

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Practice Problems

1. Find the LCM of 5 and 7.
2. Find the LCM of 10 and 15.
3. Find the LCM of 14 and 21.

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Visual Representation

Example: LCM of 4 and 6

1. Listing Multiples:

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

2. Common Multiples:

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

3. Prime Factorisation:

4 = 2²
6 = 2 × 3
LCM = 2² × 3 = 12

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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.

Learn with an example

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