What Is The Lcm Of 6, 7, And 9? Finding The Least Common Multiple

Let’s find the least common multiple (LCM) of 6, 7, 9, and 12.

The least common multiple is the smallest number that is divisible by all the numbers in the set. To find the LCM, we can use the prime factorization method.

First, let’s find the prime factorization of each number:

* 6 = 2 x 3
* 7 = 7
* 9 = 3 x 3
* 12 = 2 x 2 x 3

Now, to find the LCM, we need to take the highest power of each prime factor that appears in any of the factorizations.

* The highest power of 2 is 2 x 2 (from the factorization of 12).
* The highest power of 3 is 3 x 3 (from the factorization of 9).
* The highest power of 7 is 7 (from the factorization of 7).

Therefore, the LCM of 6, 7, 9, and 12 is 2 x 2 x 3 x 3 x 7 = 504.

This means that 504 is the smallest number that is divisible by 6, 7, 9, and 12.

Let’s break this down further:

Think of the LCM as the “common ground” where all the numbers meet. Imagine you have 6 cookies, 7 candies, 9 chocolates, and 12 lollipops. You want to divide them into equal groups so that each group has the same number of each type of treat. The LCM tells you the smallest number of groups you can make where everyone gets a fair share.

In this case, the LCM is 504. This means that you can divide your treats into 504 groups, and each group will have:

* 84 cookies (504 / 6 = 84)
* 72 candies (504 / 7 = 72)
* 56 chocolates (504 / 9 = 56)
* 42 lollipops (504 / 12 = 42)

So, the LCM helps you figure out the smallest possible way to divide things into equal groups, ensuring that everyone gets a fair share.

The least common multiple (LCM) of 7, 9, and 7 is 63. This means that 63 is the smallest number that is divisible by all three numbers. In simple terms, the LCM is the smallest number that all three numbers can divide into evenly.

Let’s break down how we arrive at 63 as the LCM.

1. Identify the Prime Factors: We first need to find the prime factors of each number. Prime factors are the numbers that divide into the original number without leaving a remainder.
* 7 is a prime number, meaning it’s only divisible by 1 and itself.
* 9 is divisible by 3 and 3, so its prime factorization is 3 x 3.

2. Find the Common and Uncommon Factors: Notice that 7 is a prime factor in our list, but it’s not a factor of 9. We also see that 3 is a factor of 9, but not of 7.

3. Multiply the Highest Powers of all Prime Factors: To get the LCM, we multiply the highest powers of all prime factors. The highest power of 7 is 1 (from 7 itself). The highest power of 3 is 2 (from 9, which is 3 x 3).
* 7¹ x 3² = 7 x 9 = 63.

This is why 63 is the LCM of 7, 9, and 7. It’s the smallest number that can be divided evenly by all three numbers.

It’s important to note that since 7 appears twice in the original list, we don’t need to include it twice in our calculations. The LCM only considers the highest power of each prime factor, even if it appears multiple times in the original list. This is because the highest power of a factor is the smallest number that all instances of that factor will divide into.

Let’s find the least common multiple (LCM) of 6, 7, 9, and 3. The least common multiple is the smallest number that is a multiple of all the numbers in the set. Here’s how we can find it:

1. Prime Factorization: Break down each number into its prime factors:
* 6 = 2 x 3
* 7 = 7
* 9 = 3 x 3
* 3 = 3
2. Identify Common and Unique Factors:
Common Factors: 3
Unique Factors: 2, 7, 3 (from the 9)
3. Multiply the Highest Powers: Multiply the highest powers of all the prime factors identified: 2 x 3² x 7 = 126

Therefore, the least common multiple of 6, 7, 9, and 3 is 126.

Understanding the LCM

The least common multiple is a useful concept in many areas, such as:

Fractions: When adding or subtracting fractions, you need to find a common denominator. This common denominator is the least common multiple of the original denominators.
Scheduling: Imagine you have three tasks that repeat on different cycles: Task A every 6 days, Task B every 7 days, and Task C every 9 days. The least common multiple (126) tells you how many days it will take for all three tasks to coincide again.
Music: In music, the least common multiple can be used to determine the length of a musical phrase or the number of beats in a bar.

The least common multiple is a fundamental concept in mathematics and helps us understand the relationships between numbers.

The least common multiple (LCM) of 6 and 9 is 18.

Let’s break down how we arrive at that answer. The LCM is the smallest number that is a multiple of both 6 and 9.

Here’s a simple way to find the LCM:

1. List the multiples of each number:
* Multiples of 6: 6, 12, 18, 24, 30…
* Multiples of 9: 9, 18, 27, 36…

2. Identify the smallest common multiple: You’ll notice that 18 appears in both lists, making it the LCM of 6 and 9.

Understanding the Importance of LCM

The concept of LCM plays a crucial role in various mathematical and practical applications.

Fractions: When adding or subtracting fractions with different denominators, finding the LCM helps determine the least common denominator. This is essential for simplifying calculations.

Scheduling: Imagine two events happening on a recurring schedule. Finding the LCM helps determine when they will occur simultaneously. For example, if one event happens every 6 days and another every 9 days, they will both happen on the 18th day.

Measurement: In scenarios involving units of measurement, the LCM can help find the smallest common unit. For instance, if you need to measure lengths in both inches and centimeters, the LCM might help determine the most appropriate unit.

Algebra: The LCM finds use in simplifying algebraic expressions, particularly when working with fractions.

We’re going to find the Least Common Multiple (LCM) of 6, 7, and 9. The LCM is the smallest number that is a multiple of all three numbers.

Here’s how to find the LCM:

List out the multiples of each number:
* Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 126, …
* Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, …
* Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, …

Identify the smallest number that appears in all three lists: This is 126.

Therefore, the LCM of 6, 7, and 9 is 126.

Understanding the LCM

The LCM is a fundamental concept in mathematics with practical applications in various fields. Think of it like finding the common ground for different rhythms or cycles.

For instance, imagine you have three clocks: one that ticks every 6 seconds, another that ticks every 7 seconds, and a third that ticks every 9 seconds. If you start them all ticking at the same time, when will they all tick together again? The answer is after 126 seconds. This is because 126 is the smallest number divisible by 6, 7, and 9.

The LCM helps us find the shortest period where repeating events will synchronize. This principle applies to many real-world scenarios, such as:

Scheduling: If you have a task that takes 6 hours, another that takes 7 hours, and a third that takes 9 hours, the LCM will help you determine when you can complete all three tasks together.
Music: In music, the LCM helps to find the common denominator of different rhythms or time signatures, ensuring that they harmonize.
Engineering: In engineering, the LCM is used to design systems where multiple components operate at different frequencies, but need to be synchronized.

The LCM is a powerful tool that simplifies understanding and coordinating recurring events or processes in various contexts.

The LCM of 6 and 7 is 42. The LCM is the smallest number that is a multiple of both 6 and 7. To find the LCM of 6 and 7, we can list out the multiples of each number and find the smallest number that appears in both lists.

Here are the first few multiples of 6 and 7:

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48…
Multiples of 7: 7, 14, 21, 28, 35, 42, 49…

As you can see, 42 is the smallest number that appears in both lists, so it’s the LCM of 6 and 7.

Understanding LCM

The LCM (Least Common Multiple) is a fundamental concept in mathematics, particularly when dealing with fractions and working with numbers that share common factors. It plays a crucial role in simplifying fractions, finding common denominators, and solving various mathematical problems.

Why is the LCM important?

The LCM is useful because it helps us find the smallest common denominator when adding or subtracting fractions with different denominators. To add or subtract fractions, they need to have the same denominator. Finding the LCM of the denominators allows us to create equivalent fractions with a common denominator, making the addition or subtraction process easier.

For instance, let’s say we want to add the fractions 1/6 and 1/7. The LCM of 6 and 7 is 42. To get a common denominator of 42, we multiply the numerator and denominator of the first fraction by 7 and the numerator and denominator of the second fraction by 6. This gives us:

* 1/6 * 7/7 = 7/42
* 1/7 * 6/6 = 6/42

Now, with a common denominator of 42, we can add the fractions:

* 7/42 + 6/42 = 13/42

In Summary

The LCM is a valuable tool in mathematics, simplifying operations with fractions and enabling us to work effectively with numbers that share common factors. It is a concept that finds application in various areas of mathematics, making it a crucial element in our understanding of numbers and their relationships.

The least common multiple of 9, 8, 7 and 6 is 504.

Let’s break down how we find this answer. The least common multiple (LCM) is the smallest number that is a multiple of all the numbers in a set. To find the LCM, we can follow these steps:

1. Prime factorization: Break each number down into its prime factors.
– 9 = 3 x 3
– 8 = 2 x 2 x 2
– 7 = 7 (already prime)
– 6 = 2 x 3
2. Identify all the prime factors: We have 2, 3, and 7.
3. Take the highest power of each prime factor:
– 2 appears a maximum of three times (in 8)
– 3 appears a maximum of two times (in 9)
– 7 appears once
4. Multiply the highest powers together: 2³ x 3² x 7 = 8 x 9 x 7 = 504

Therefore, the LCM of 7, 6, 9, and 8 is 504. This means that 504 is the smallest number that can be divided evenly by all four numbers.

The LCM of 6, 7, and 9 is 126. LCM stands for Least Common Multiple or Lowest Common Multiple. It’s the smallest number that is a multiple of all the numbers you’re working with.

Let’s break down how to find the LCM of 6, 7, and 9:

Prime Factorization: Start by finding the prime factors of each number:
6: 2 x 3
7: 7 (7 is a prime number)
9: 3 x 3
Identify Common and Unique Factors: Look for the factors that appear in all the numbers, and then note any unique factors. In this case, we have:
Common Factors: 3
Unique Factors: 2, 7, 3
Calculate the LCM: Multiply the highest power of each factor together:
* 2 x 3² x 7 = 126

Here’s a more intuitive way to think about LCM:

Imagine you have three groups of objects: 6 apples, 7 oranges, and 9 bananas. You want to arrange them into equal rows, with the same number of each fruit in each row. The smallest number of rows you could have is 126. This means you’d have 21 rows of apples, 18 rows of oranges, and 14 rows of bananas.

Understanding the LCM is essential in various mathematical concepts like:

Fractions: Finding the LCM helps in adding and subtracting fractions with different denominators.
Algebra: It’s used when dealing with algebraic expressions involving fractions and multiples.
Real-World Problems: LCM is used to solve real-world problems, such as figuring out when two events will happen at the same time or determining the least amount of material needed to complete a task.

Let’s find the LCM of 6, 7, and 9 using the division method.

Here’s how we do it:

1. Write the numbers 6, 7, and 9 in a row.
2. Find the smallest prime number that divides at least one of the numbers. In this case, it’s 2. Divide the numbers divisible by 2 by 2, and bring down the numbers that are not divisible.
3. Repeat Step 2 with the new numbers in the row. Continue this process until you get only 1s in the last row.
4. Multiply the prime numbers you used to divide. This product is the LCM of 6, 7, and 9.

Let’s break this down for the numbers 6, 7, and 9:

“`
2 | 6 7 9
3 7 9
3 | 3 7 9
1 7 3
7 | 1 7 3
1 1 3
3 | 1 1 3
1 1 1
“`

Here’s how we calculated the LCM:

– We divided by 2 once.
– We divided by 3 twice.
– We divided by 7 once.

Therefore, the LCM of 6, 7, and 9 is 2 * 3 * 3 * 7 = 126.

Understanding the Division Method

The division method is based on the idea of finding the prime factorization of each number.

– Prime Factorization: Breaking down a number into its prime factors (prime numbers that multiply together to give the original number). For example, the prime factorization of 6 is 2 * 3.
– LCM: The Least Common Multiple (LCM) is the smallest number that is a multiple of all the given numbers. In essence, it’s the smallest number that can be divided by each of the original numbers without leaving a remainder.

By repeatedly dividing by prime numbers, we are essentially finding the prime factors of each of the original numbers. The LCM is the product of the highest powers of all the prime factors present in the numbers.

Example:

Let’s look at the LCM of 6, 7, and 9 again:

– Prime factorization of 6: 2 * 3
– Prime factorization of 7: 7
– Prime factorization of 9: 3 * 3

The LCM is the product of the highest powers of all the prime factors: 2 * 3 * 3 * 7 = 126.

Let’s find the least common multiple (LCM) of 6, 7, and 9 using prime factorization.

To find the LCM, we follow these steps:

1. Prime Factorization: Break down each number into its prime factors.
* 6 = 2 x 3
* 7 = 7 (already a prime number)
* 9 = 3 x 3

2. Identify the Highest Powers: Look at each prime factor and determine the highest power it appears in any of the numbers.
2 appears with the highest power of 1 (in 6)
3 appears with the highest power of 2 (in 9)
7 appears with the highest power of 1 (in 7)

3. Multiply the Prime Factors: Multiply the prime factors raised to their highest powers.
* 2¹ x 3² x 7¹ = 2 x 9 x 7 = 126

Therefore, the LCM of 6, 7, and 9 is 126.

Understanding the Concept of Least Common Multiple (LCM)

The LCM is the smallest positive integer that is a multiple of all the given numbers. In other words, it’s the smallest number that all the numbers divide into evenly.

Prime Factorization and LCM

Prime factorization is a powerful tool for finding the LCM because it breaks down numbers into their fundamental building blocks. By identifying the highest powers of each prime factor, we ensure that our final result is a multiple of all the original numbers.

Example

Imagine you have three pieces of ribbon: one 6 inches long, another 7 inches long, and a third 9 inches long. You want to cut all three ribbons into equal pieces without any leftover ribbon. The LCM (126 inches) represents the longest possible length of the equal pieces you can cut. This means you can cut each ribbon into 21 pieces, 18 pieces, and 14 pieces respectively, with no waste.

Let’s find the smallest common multiple of 6, 7, and 9. We’ll use a method called prime factorization.

First, we break down each number into its prime factors:

6 = 2 x 3
7 = 7 (7 is a prime number)
9 = 3 x 3

Now, we take the highest power of each prime factor that appears in any of the numbers:

2¹ (from 6)
3² (from 9)
7¹ (from 7)

Finally, we multiply these highest powers together:

* 2¹ x 3² x 7¹ = 2 x 9 x 7 = 126

Therefore, the smallest common multiple (LCM) of 6, 7, and 9 is 126.

Think of it this way: The LCM is the smallest number that is divisible by all three of our original numbers. You can verify this by dividing 126 by 6, 7, and 9 – you’ll get a whole number in each case.

Here’s why prime factorization works:

Prime numbers are the building blocks of all other numbers. By breaking our numbers down into prime factors, we’re essentially looking at the “ingredients” that make up each number.
The LCM must contain all the prime factors of each of the original numbers. This ensures that it’s divisible by each original number.
We use the highest power of each prime factor. This ensures that the LCM is the smallest possible number that satisfies the condition of being divisible by all the original numbers.

Let’s illustrate this with an example:

Imagine you have three groups of people, each wanting to share a cake evenly. Group A wants to divide their cake into 6 slices, Group B wants 7 slices, and Group C wants 9 slices. To make sure everyone gets an even share, we need to find the smallest number of slices that is divisible by 6, 7, and 9. This is where the LCM comes in – it tells us we need to cut the cake into 126 slices!

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