Lesson 04 Comparing the GCD and the LCM - SimpleStep Learning

SimpleStep Learning
17 Aug 201601:48

Summary

TLDRThis lesson clarifies the concepts of greatest common divisor (GCD) and least common multiple (LCM). The GCD is the highest number that divides two numbers without a remainder, exemplified by the GCD of 4 and 6 being 2. Conversely, the LCM is the smallest number that both numbers can divide into, like the LCM of 4 and 6 being 12. The lesson illustrates these with examples, including 6 and 9, and concludes with a challenge to find the GCD and LCM of 5 and 10, which are 5 and 10, respectively. It highlights that the GCD is not greater than the smaller number, while the LCM is not less than the larger number.

Takeaways

  • 📚 The GCD (Greatest Common Divisor) is the largest number that divides two given numbers without leaving a remainder.
  • 🔱 The LCM (Least Common Multiple) is the smallest number that is a multiple of two given numbers.
  • 🌰 An example given is the GCD of 4 and 6, which is 2, as it's the largest number that divides both 4 and 6.
  • 📈 The LCM of 4 and 6 is 12, as it's the smallest number that both 4 and 6 can divide into without a remainder.
  • 👀 The GCD of 6 and 9 is 3, highlighting that it's the largest factor common to both numbers.
  • 🔄 The LCM of 6 and 9 is 18, showing it's the smallest number that is a multiple of both 6 and 9.
  • 💡 The GCD of 5 and 10 is 5, demonstrating that if one number is a factor of the other, it's the GCD.
  • 🔑 The LCM of 5 and 10 is 10, indicating that if one number is a multiple of the other, it's the LCM.
  • 📉 It's noted that the GCD is always less than or equal to the smaller number in the pair.
  • 📈 Conversely, the LCM is always greater than or equal to the larger number in the pair.

Q & A

  • What is the GCD (Greatest Common Divisor)?

    -The GCD is the greatest number that divides two or more numbers without leaving a remainder. It is the largest factor that is common to all the numbers in a given set.

  • How do you find the GCD of 4 and 6?

    -The GCD of 4 and 6 is 2. This is because 2 is the largest number that is a factor of both 4 and 6.

  • What is the LCM (Least Common Multiple)?

    -The LCM is the smallest number that is a multiple of two or more numbers. It is the smallest number that all the numbers in a set can divide into without leaving a remainder.

  • Can you provide the LCM of 4 and 6 as an example?

    -The LCM of 4 and 6 is 12. This is because 12 is the smallest number that is a multiple of both 4 and 6.

  • What is the GCD of 6 and 9?

    -The GCD of 6 and 9 is 3. This is because 3 is the largest number that is a factor of both 6 and 9.

  • How do you calculate the LCM of 6 and 9?

    -The LCM of 6 and 9 is 18. This is because 18 is the smallest number that is a multiple of both 6 and 9.

  • What is the GCD of 5 and 10, and why?

    -The GCD of 5 and 10 is 5. This is because 5 is a factor of 10, making it the greatest common divisor of the two numbers.

  • What is the LCM of 5 and 10, and how is it determined?

    -The LCM of 5 and 10 is 10. This is because 10 is the smallest number that is a multiple of both 5 and 10.

  • Is there a relationship between the GCD and LCM of two numbers?

    -Yes, the product of the GCD and LCM of two numbers is equal to the product of the numbers themselves. This relationship is often used in calculations involving divisors and multiples.

  • Why is the GCD always less than or equal to the smaller number in a pair?

    -The GCD is the largest common factor, and it cannot be larger than the smallest number in the pair because it must be a factor of both numbers, and the smaller number is the limiting factor.

  • Why is the LCM always greater than or equal to the larger number in a pair?

    -The LCM is the smallest common multiple, and it must be at least as large as the largest number in the pair because it must be a multiple of both numbers, and the larger number sets the minimum for the smallest common multiple.

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GCDLCMMathematicsEducationNumber TheoryDivisibilityMultiplesFactorsLearningTutorial
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