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.

Outlines

00:00

📘 Understanding GCD and LCM

This paragraph introduces the concepts of the Greatest Common Divisor (GCD) and the Least Common Multiple (LCM). It explains that the GCD is the largest number that divides two given numbers without leaving a remainder, using the example of 4 and 6, which have a GCD of 2. Conversely, the LCM is the smallest number that is a multiple of both given numbers, with 4 and 6 having an LCM of 12. The paragraph further clarifies these concepts by providing another example, where the GCD of 6 and 9 is 3, and their LCM is 18. It concludes with an example of the GCD and LCM of 5 and 10, which are 5 and 10, respectively, and highlights that the GCD is always less than or equal to the smaller number, while the LCM is always greater than or equal to the larger number.

Keywords

💡GCD (Greatest Common Divisor)

The GCD is the largest number that divides two or more numbers without leaving a remainder. In the context of the video, it is used to distinguish between the common factors of two numbers. For instance, the GCD of 4 and 6 is 2, as it is the highest number that divides both 4 and 6. The concept is crucial in understanding the relationship between numbers and their factors.

💡LCM (Least Common Multiple)

The LCM is the smallest number that is a multiple of two or more numbers. It is the inverse concept of GCD and is used to identify the smallest number that can be divided by a set of numbers without a remainder. In the video, the LCM of 4 and 6 is given as 12, which is the smallest number that both 4 and 6 can divide into without leaving a remainder. This concept helps in understanding multiples and the commonality between different numbers.

💡Common Divisor

A common divisor is a number that divides two or more numbers without leaving a remainder. The video script uses this term to explain the concept of finding the greatest common divisor (GCD). For example, 2 is a common divisor of 4 and 6, and since it is the greatest such number, it is also their GCD.

💡Common Multiple

A common multiple is a number that is a multiple of two or more numbers. The video script uses this term to explain the concept of finding the least common multiple (LCM). For example, 12 is a common multiple of 4 and 6, and since it is the smallest such number, it is their LCM.

💡Factor

A factor is a number that divides another number without leaving a remainder. The video script uses the term 'factor' to explain how to determine the GCD, as the GCD is the greatest factor that is common to the numbers in question. For example, 5 is a factor of 10, and since it is the greatest common factor of both 5 and 10, it is their GCD.

💡Multiple

A multiple is a number that is the product of another number and an integer. The video script uses the term 'multiple' to explain how to determine the LCM, as the LCM is the smallest multiple common to the numbers in question. For example, 10 is a multiple of both 5 and 10, and since it is the smallest such number, it is their LCM.

💡Divisibility

Divisibility refers to the property of a number being evenly divisible by another number without leaving a remainder. The video script discusses divisibility in the context of finding the GCD and LCM, as these concepts are fundamentally about identifying numbers that divide other numbers without remainders.

💡Divisor

A divisor is a number that divides another number. The term is used in the video script to explain the process of finding the GCD, where one identifies the divisors of each number and then determines the greatest among them that is common to both.

💡Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors. While not explicitly mentioned in the script, this concept is related to finding the GCD and LCM, as understanding the prime factors of numbers can help in determining their common divisors and multiples.

💡Integer

An integer is a whole number, positive, negative, or zero, without a fractional component. The video script discusses integers in the context of multiples and factors, as the GCD and LCM are typically integers. For example, the GCD of 6 and 9 is 3, which is an integer, and the LCM of 6 and 9 is 18, also an integer.

💡Smallest Number

In the context of the video, the 'smallest number' refers to the LCM, which is the least common multiple of two or more numbers. The script uses this term to emphasize that the LCM is the smallest number that is a multiple of the given numbers, such as 12 being the smallest number that is a multiple of both 4 and 6.

💡Greatest Number

In the context of the video, the 'greatest number' refers to the GCD, which is the greatest common divisor of two or more numbers. The script uses this term to highlight that the GCD is the largest number that is a divisor of the given numbers, such as 5 being the greatest number that is a divisor of both 5 and 10.

Highlights

The gcd (greatest common divisor) is the largest number that divides two numbers without leaving a remainder.

The LCM (least common multiple) is the smallest number that is a multiple of two numbers.

For example, the gcd of 4 and 6 is 2, as it is the greatest number that divides both.

The LCM of 4 and 6 is 12, as it is the smallest number that is a multiple of both.

The gcd of 6 and 9 is 3, as it is the greatest number that is a factor of both.

The LCM of 6 and 9 is 18, as it is the smallest number that is a multiple of both.

The gcd of 5 and 10 is 5, since 5 is a factor of 10 and thus the greatest common divisor.

The LCM of 5 and 10 is 10, as it is the smallest number that is a multiple of both.

The gcd of two numbers is always less than or equal to the smaller number.

The LCM of two numbers is always greater than or equal to the larger number.

Understanding gcd and LCM is crucial for solving problems involving the relationship between numbers.

The concepts of gcd and LCM are fundamental in number theory and have practical applications.

The gcd can be found by identifying the common factors of two numbers.

The LCM can be calculated by finding the smallest number that includes the multiples of both numbers.

The relationship between gcd and LCM can be used to simplify fractions.

In cases where one number is a multiple of the other, the gcd is the smaller number.

When one number is a multiple of another, the LCM is the larger number.

The gcd and LCM are interconnected, as the product of the two for any two numbers is equal to the product of the numbers themselves.

These mathematical concepts are essential for various mathematical operations and algorithms.

Transcripts

Lesson Four distinguishing the gcd and

the

LCM you may find the gcd and the LCM a

little confusing at first let's look at

both of these Concepts to help you keep

them

clear remember that the gcd is the

greatest common divisor for example the

gcd of four and six is two because two

is the greatest number that's a of both

four and six in contrast the LCM is the

least common multiple the LCM of 4 and 6

is 12 because 12 is the smallest number

that's a multiple of both four and

six as another example the gcd of 6 and

9 is three because three is the greatest

number that's a factor of both 6 and 9

and the LCM of 6 and 9 is 18 because 18

is the smallest number that's a multiple

of both

numbers here's one final example what is

the gcd of five and 10 the answer is

five because five is a factor of 10 so

it's the greatest common divisor of five

and 10 now can you find the LCM of 5 and

10 it's 10 because 10 is the smallest

number that's a multiple of both

numbers notice that the gcd of two

numbers is always less than or equal to

the lower number while the LCM is always

greater than or equal to the greater

number