# Exploring Factors of 54: Find Prime Factors, Pairs, Common & Distinct Factors of 54

June 18, 2023

Are you finding factors and their properties puzzling? Don’t worry, we’ve got you covered! In this comprehensive article, we’ll dive into the fascinating world of numbers, specifically focusing on the number 54. We understand that students often face challenges when dealing with factors, which is why we’ve compiled all the information you need to demystify the factors of 54.

Let’s begin by exploring the fundamental question: “What are the factors of 54?” We’ll unravel the factors that evenly divide 54 and provide you with a clear understanding of how they work. Next, we’ll delve into the prime factors of 54, uncovering the key prime numbers that contribute to its factorization. By understanding the prime factors, you’ll gain insights into the unique components that make up 54.

Moving on, we’ll explore the factors of 54 in pairs, revealing intriguing combinations that result in this number. We’ll showcase how factors come together in pairs to create the whole. Additionally, we’ll investigate the common factors of 54, which are shared by 54 and other numbers. This exploration will help you recognize the patterns and relationships between 54 and its common factors. Lastly, we’ll delve into the distinct factors of 54, highlighting the factors that make it stand out from other numbers.

## What are the Factors of 54?

Factors are the numbers that divide another number evenly without leaving a remainder. In the case of 54, the factors are the numbers that divide 54 without any remainder. The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54. These numbers, when multiplied together, result in 54.

### How to Find Factors of 54

Here are the step-by-step instructions to find the factors of 54:

Step 1: Start by listing the number 1 as a factor, as every number is divisible by 1.

Step 2: Divide 54 by the number 2. Since 54 is evenly divisible by 2, write down 2 as a factor and divide 54 by 2, resulting in 27.

Step 3: Divide 27 by the next prime number, which is 3. Since 27 is evenly divisible by 3, write down 3 as a factor and divide 27 by 3, resulting in 9.

Step 4: Continue dividing 9 by the next prime number, which is again 3. Since 9 is evenly divisible by 3, write down 3 as a factor and divide 9 by 3, resulting in 3.

Step 5: Finally, write down 3 as a factor, as it is the remaining quotient.

Therefore, the factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.

## What are the Prime Factors of 54?

Prime factors are the factors that are prime numbers. In the case of 54, the prime factors are the prime numbers that divide 54 exactly. The prime factors of 54 are 2 and 3. These prime factors, when multiplied together, yield the original number, 54.

### How to Find the Prime Factors of 54

To find the prime factors of 54, we divide them by prime numbers until we cannot divide them further. By following a step-by-step process, we find the prime factors of 54: 2 and 3. These prime factors can also be organized in a table format, providing a concise overview.

To find the prime factors of 54, follow these steps:

Step 1: Start with the number 2, which is the smallest prime number.

Step 2: Divide 54 by 2. Since 54 is evenly divisible by 2, write down 2 as a prime factor and divide 54 by 2, resulting in 27.

Step 3: Divide 27 by the next prime number, which is 3. Since 27 is evenly divisible by 3, write down 3 as a prime factor and divide 27 by 3, resulting in 9.

Step 4: Divide 9 by 3 again, since it is still divisible by 3. Write down 3 as another prime factor and divide 9 by 3, resulting in 3.

Step 5: Write down 3 as the final prime factor, as it is the remaining quotient.

Therefore, the prime factors of 54 are 2, 3, and 3.

## Factors of 54 in Pairs

Factors in pairs are the pairs of numbers that multiply together to give the original number. For 54, the factors can be paired as (1, 54), (2, 27), (3, 18), and (6, 9). These pairs demonstrate the unique combinations of factors that result in product 54.

Positive Pairs:

Negative Pairs:

In the positive pairs table, each pair consists of a factor of 54 and its corresponding factor that multiplies to give the product of 54. For example, 2 and 27 are factors of 54, and their product is also 54.

In the negative pairs table, each pair consists of a negative factor of 54 and its corresponding negative factor that multiplies to give the product of 54. The product remains the same as in the positive pairs table because multiplying two negative numbers gives a positive result.

These pairs of factors help us understand the different ways in which 54 can be divided into two numbers whose product is 54.

## What are the Common Factors of 54?

Common factors are the factors that two or more numbers have in common. For example, the common factors of 54 and 72 are 1, 2, 3, 6, 9, 18, and 54. These numbers divide both 54 and 72 evenly, demonstrating their shared factors.

Here are 10 examples of common factors of 54 with other numbers:

1. Common factors of 54 and 18:
1, 2, 3, 6, 9, 18
2. Common factors of 54 and 27:
1, 3, 9, 27
3. Common factors of 54 and 36:
1, 2, 3, 6, 9, 18, 36
4. Common factors of 54 and 45:
1, 3, 9
5. Common factors of 54 and 72:
1, 2, 3, 6, 9, 18
6. Common factors of 54 and 81:
1, 3, 9
7. Common factors of 54 and 90:
1, 2, 3, 6, 9, 18
8. Common factors of 54 and 108:
1, 2, 3, 6, 9, 18, 27, 54
9. Common factors of 54 and 135:
1, 3, 9
10. Common factors of 54 and 162:
1, 2, 3, 6, 9, 18, 27, 54

## What are the Distinct Factors of 54?

Distinct factors refer to the unique factors of a number. In the case of 54, the distinct factors are the factors that are not repeated. The distinct factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54, excluding any duplicates.

Here are 10 examples of distinct factors of 54 with other numbers:

1. Distinct factors of 54 and 20:
1, 2, 5, 10
2. Distinct factors of 54 and 30:
1, 2, 3, 5, 6, 9, 10, 15
3. Distinct factors of 54 and 40:
1, 2, 4, 5, 8, 10, 20
4. Distinct factors of 54 and 50:
1, 2, 5, 10, 25
5. Distinct factors of 54 and 60:
1, 2, 3, 6, 9, 18, 30
6. Distinct factors of 54 and 70:
1, 2, 5, 7, 10, 14, 35
7. Distinct factors of 54 and 80:
1, 2, 4, 5, 8, 10, 16, 20, 40
8. Distinct factors of 54 and 90:
1, 2, 3, 5, 6, 9, 10, 15, 18, 30
9. Distinct factors of 54 and 100:
1, 2, 4, 5, 10, 20, 25, 50
10. Distinct factors of 54 and 110:
1, 2, 5, 10, 11, 22, 55

These examples show the factors of 54 that are distinct and do not overlap with the factors of the given number.

Conclusion: By thoroughly examining the factors of 54, we have uncovered a wealth of information about this intriguing number. From understanding the factors themselves to discovering their relationships, such as prime factors, factors in pairs, common factors, and distinct factors, we have gained valuable insights into the properties of 54. This comprehensive guide equips you with the knowledge and tools to explore and analyze factors not only for 54 but also for other numbers, enriching your understanding of the fascinating world of mathematics.

## Solved Factors of 54 with 10 Examples

1. Find the factors of 54?
The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.

2. Are 36 and 54 relatively prime? Why or why not?
No, 36 and 54 are not relatively prime because they share common factors.

3. Determine whether 54 is a perfect square. If not, explain why.
No, 54 is not a perfect square because it does not have an integer square root.

4. How many even factors does 54 have? List them.
54 has three even factors: 2, 6, and 54.

5. Is 54 divisible by 7? Why or why not?
No, 54 is not divisible by 7 because 7 does not evenly divide into 54.

6. Find the greatest common factor (GCF) of 54 and 72.
The greatest common factor (GCF) of 54 and 72 is 18.

7. Can you express 54 as a product of prime factors? If yes, provide the prime factorization.
Yes, the prime factorization of 54 is 2 * 3^3.

8. What is the sum of the factors of 54?
The sum of the factors of 54 is 117.

9. Which numbers are both factors of 54 and multiples of 3?
The numbers 3, 6, 9, 18, 27, and 54 are both factors of 54 and multiples of 3.

10. Find the prime factors of 54
The prime factors of 54 are 2, 3, and 3.

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