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G.4 Least common multiple

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What is the least common multiple?

The least common multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of each of the given numbers.

Examples:
  • The LCM of 4 and 6 is 12
  • The LCM of 3 and 5 is 15
  • The LCM of 2, 3, and 4 is 12
Note

Think of the LCM as the smallest number where all your original numbers can "fit" evenly as factors.

How to find the LCM by listing multiples

One way to find the LCM is to list the multiples of each number and identify the smallest multiple that appears in all lists.

Example: Find the LCM of 4 and 6
  • Multiples of 4: 4, 8, 12, 16, 20, 24...
  • Multiples of 6: 6, 12, 18, 24, 30...
  • The smallest common multiple is 12
Note

This method works well for smaller numbers but can become time-consuming with larger numbers.

How to find the LCM using prime factorization

For larger numbers, we can use prime factorization to find the LCM more efficiently. Write each number as a product of its prime factors, then multiply the highest power of each prime factor together.

Example: Find the LCM of 12 and 18
  • Prime factors of 12: 2 × 2 × 3 = 22 × 31
  • Prime factors of 18: 2 × 3 × 3 = 21 × 32
  • Highest power of 2: 22
  • Highest power of 3: 32
  • LCM = 22 × 32 = 4 × 9 = 36
Note

Make sure to include every prime factor that appears in any of the numbers, using the highest exponent found.

Real-world applications of LCM

The least common multiple helps us solve problems where we need to find when different cycles or patterns will align.

Examples:
  • Two buses arrive at a station every 15 minutes and 20 minutes. If they arrive together at 8:00 AM, the LCM of 15 and 20 (which is 60) tells us they will next arrive together 60 minutes later at 9:00 AM.
  • If you water plants every 4 days and fertilize every 10 days, the LCM of 4 and 10 (which is 20) tells you that both tasks will occur on the same day every 20 days.
Note

Look for problems that involve finding when repeating events will happen at the same time - these often use LCM.

Special cases and relationships

There are special relationships between numbers that can help you find LCM more quickly.

Important relationships:
  • If one number is a multiple of the other, the LCM is the larger number (LCM of 4 and 12 is 12)
  • For two prime numbers, the LCM is their product (LCM of 5 and 7 is 35)
  • For consecutive numbers, the LCM is usually their product (LCM of 3 and 4 is 12)
Note

Recognizing these patterns can save you time when solving problems.

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