Lesson 04 Comparing the GCD and the LCM - SimpleStep Learning

Summary

This 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.