Table of Contents

Lowest common multiples Level 7

Introduction

Have you ever tried to find a common time to meet with friends who have different schedules? That’s similar to what we do when we find the lowest common multiple (LCM) of numbers! Understanding LCM helps us solve problems involving fractions, ratios, and even scheduling events. Let’s dive into the world of LCM and discover its importance in mathematics and real life.

Definition and Concept

The lowest common multiple (LCM) of two or more numbers is the smallest multiple that is evenly divisible by all the numbers. For instance, the LCM of 4 and 5 is 20, because 20 is the smallest number that both 4 and 5 can divide into without leaving a remainder.

Relevance:

  • Mathematics: LCM is essential for adding and subtracting fractions.
  • Real-world applications: Helps in scheduling, planning events, and solving problems involving repeated cycles.

Historical Context or Origin​

The concept of multiples has been used since ancient times. The Greeks and the Chinese developed methods for finding LCMs, which were essential for their mathematical calculations. The systematic approach to finding LCMs evolved with the development of number theory, particularly during the Renaissance period.

Understanding the Problem

To find the LCM of two or more numbers, we can use two primary methods: listing multiples and prime factorization. Let’s break down these methods:

Methods to Solve the Problem with different types of problems​

Method 1: Listing Multiples

  • List the multiples of each number.
  • Identify the smallest multiple that appears in all lists.

Example:
Find the LCM of 3 and 4.
Multiples of 3: 3, 6, 9, 12, 15…
Multiples of 4: 4, 8, 12, 16…
The LCM is 12.

Method 2: Prime Factorization

  • Factor each number into its prime factors.
  • Take the highest power of each prime factor that appears in any of the factorizations.
  • Multiply these together to get the LCM.

Example:
Find the LCM of 6 and 8.
6 = 21 × 31
8 = 23
LCM = 23 × 31 = 24.

Exceptions and Special Cases​

  • One of the Numbers is Zero: The LCM of any number and zero is always zero.
  • Identical Numbers: The LCM of a number with itself is the number itself (e.g., LCM of 5 and 5 is 5).

    • Step-by-Step Practice​

    Problem 1: Find the LCM of 12 and 15.

    Solution:

  • Multiples of 12: 12, 24, 36, 48, 60…
  • Multiples of 15: 15, 30, 45, 60…
  • The LCM is 60.

    • Problem 2: Find the LCM of 9 and 21.

    Solution:

    1. Prime factorization: 9 = 32, 21 = 31 × 71.
    2. LCM = 32 × 71 = 63.

    Examples and Variations

    Easy Example:

    • Problem: Find the LCM of 2 and 5.
    • Solution:
      • Multiples of 2: 2, 4, 6, 8, 10…
      • Multiples of 5: 5, 10, 15…
      • LCM is 10.

    Moderate Example:

    • Problem: Find the LCM of 4 and 6.
    • Solution:
      • Multiples of 4: 4, 8, 12, 16…
      • Multiples of 6: 6, 12, 18…
      • LCM is 12.

    Advanced Example:

    • Problem: Find the LCM of 10, 15, and 20.
    • Solution:
      • Prime factorization: 10 = 21 × 51, 15 = 31 × 51, 20 = 22 × 51.
      • LCM = 22 × 31 × 51 = 60.

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    Common Mistakes and Pitfalls

    • Forgetting to include all prime factors when using the prime factorization method.
    • Not checking the multiples carefully when using the listing method.
    • Confusing LCM with greatest common divisor (GCD).

    Tips and Tricks for Efficiency

    • When using prime factorization, write down the prime factors clearly to avoid confusion.
    • For larger numbers, the prime factorization method can be quicker than listing multiples.
    • Remember that the LCM is always greater than or equal to the largest number in the set.

    Real life application

    • Scheduling: Finding times for events that repeat every few days or weeks.
    • Cooking: Adjusting recipes that serve different numbers of people.
    • Sports: Planning tournaments where teams play in rounds.

    FAQ's

    The LCM of 1 and any number is that number itself, since any number is a multiple of 1.
    No, the LCM is always equal to or larger than the largest number in the set.
    You can find the LCM of multiple numbers by finding the LCM of two numbers at a time or using prime factorization for all numbers.
    Using prime factorization is often more efficient for larger numbers or when multiples are hard to list.
    Yes, the product of the LCM and GCD of two numbers equals the product of the numbers themselves.

    Conclusion

    Understanding the lowest common multiple is a vital skill in mathematics that helps us solve various problems, especially those involving fractions and scheduling. By practicing different methods, you will become proficient in finding LCMs and applying them in real-life situations.

    References and Further Exploration

    • Khan Academy: Interactive lessons on LCM and GCD.
    • Book: Mathematics for the Nonmathematician by Morris Kline.

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