Find Factors of 21 by Prime Factorization and Factor Pairs

Let’s talk about factors—specifically, the factors of 21. If you’ve ever wondered what numbers can team up to create 21 through multiplication, you’re in the right place.

Factors are like building blocks of a number. For example, 3 and 7 are factors of 21 because when you multiply them (3 × 7), you get 21.

In this article, we will explore what all the factors of 21 are, how to find factors of 21 using simple methods like division and multiplication, and dive into concepts like prime factorization, factor trees, and factor in pairs. We’ll also look at examples and discuss common mistakes to help you master the topic with confidence.

What are the factors of 21?

Factors are numbers that multiply together to give another number. The smallest factor of 21 is 1. For the number 21, its factors are the numbers that can divide it evenly with zero remainder. Let’s break it down:

• 1 × 21 = 21, so 1 and 21 are factors of 21.
• 3 × 7 = 21, so 3 and 7 are also factors of 21.

These are the only numbers that work. Therefore, the factors of 21 are 1, 3, 7, and 21. These numbers are special because they are the building blocks of 21 in multiplication. Now, let’s move on to how we can find these factors using different methods.

Understanding the Factors of 21

Understanding the factors of 21 is essential in mathematics, as it helps in various mathematical operations such as division, multiplication, and finding the greatest common factor (GCF) or least common multiple (LCM) of two numbers. The factors of 21 are the numbers that divide 21 exactly without leaving a remainder. These factors can be positive or negative integers.

For instance, the positive factors of 21 are 1, 3, 7, and 21. These numbers can divide 21 without leaving any remainder, making them the building blocks of 21. But factors aren’t limited to positive numbers. Negative integers can also be factors if they multiply together to give 21. For example, -1, -3, -7, and -21 are negative factors of 21 because (-1) × (-21) = 21 and (-3) × (-7) = 21.

Understanding both positive and negative factors is crucial, especially when solving equations or working with algebraic expressions. Knowing the factors of a number like 21 can simplify complex problems and make mathematical operations more manageable.

How to find the factors of 21?

There are two simple ways to find the factors of 21: the multiplication method and the division method. Pair factors are pairs of numbers that multiply to give 21. Let’s explore both.

Factors of 21 by Multiplication Method

In this method, we look for pairs of numbers that multiply together to give 21. Here’s how it works:

• Start with 1 × 21 = 21. This means 1 and 21 are factors.
• Next, try 3 × 7 = 21. This means 3 and 7 are also factors.
• Negative pair factors like (-3) and (-7) also multiply to give 21.
• There are no other whole numbers that multiply to give 21, so we stop here.

Using this method, we find the factors of 21 are 1, 3, 7, and 21.

Factors of 21 by Division Method

In this method, we divide 21 by different numbers to see if they leave no remainder. Here’s how it works:

• Divide 21 by 1: 21 ÷ 1 = 21. Since there’s no remainder, 1 and 21 are factors.
• Divide 21 by 3: 21 ÷ 3 = 7. Since there’s no remainder, 3 and 7 are factors.
• Divide 21 by 7: 21 ÷ 7 = 3. Again, no remainder, so 7 and 3 are factors.
• Divide 21 by 21: 21 ÷ 21 = 1. No remainder, so 21 and 1 are factors.

If we try dividing 21 by numbers like 2, 4, 5, or 6, they leave remainders, so they are not factors.

Using both methods, we confirm that the factors of 21 are 1, 3, 7, and 21.

Prime Factorization of 21

Prime factorization is a way of breaking down a number into its smallest building blocks—prime numbers. For example, the prime factors of 21 are 3 and 7. Prime numbers are numbers greater than 1 that can only be divided by 1 and themselves.

Let’s explore it step by step:

1. Start with the smallest prime number, which is 2. Can 21 be divided by 2? No, because 21 is an odd number.
2. Move to the next prime number, which is 3. Divide 21 by 3:
3. 21 ÷ 3 = 7.
4. Now, we have 3 and 7.
5. Check if 7 is a prime number. Since 7 can only be divided by 1 and itself, it is a prime number.

So, the prime factorization of 21 is 3 × 7. These are the prime numbers that multiply together to give 21.

Factor Tree of 21

A factor tree is a visual way to break down a number of its prime factors. It’s like drawing a tree where the branches split into smaller and smaller factors until we reach the prime numbers.

Let’s build a factor tree for 21:

1. Start with the number 21 at the top of the tree.
2. Think of two numbers that multiply to give 21. We know 3 × 7 = 21, so we draw two branches: one for 3 and one for 7.
3. Now, check if 3 and 7 can be broken down further.
   1. 3 is a prime number (it can only be divided by 1 and itself), so we stop here.
   2. 7 is also a prime number, so we stop here as well.

The factor tree for 21 looks like this:

At the bottom of the tree, we have the prime factors of : 3 and 7. This shows that 21 is made by multiplying these two numbers together.

Factor Pairs of 21

Factor pairs are two numbers that multiply together to give the original number. For 21, we can find both positive factor pairs and negative factor pairs. Let’s explore both:

Positive Factor Pairs of 21

These are pairs of positive numbers that multiply to give 21. Using the factors we already know (1, 3, 7, 21), we can list a table:

Factor Pair	Multiplication
1 × 21	1 × 21 = 21
3 × 7	3 × 7 = 21

These are the only positive pairs that work.

Negative Factor Pairs of 21

Negative factor pairs are pairs of negative numbers that multiply to give 21. Remember, multiplying two negative numbers gives a positive result. Here are the negative factor pairs:

Factor Pair	Multiplication
(-1) × (-21)	(-1) × (-21) = 21
(-3) × (-7)	(-3) × (-7) = 21

These pairs also multiply to give 21, but they use negative numbers.

Summary of Factor Pairs

Here’s a table summarizing all the factor pairs of 21:

Positive Pairs	Negative Pairs
1 × 21	(-1) × (-21)
3 × 7	(-3) × (-7)

Whether positive or negative, these pairs all multiply to give 21.

Solved Examples on Factors of 21

1.What Is the Greatest Common Factors of 21?

The Greatest Common Factor (GCF) of 21 depends on the other number you’re comparing it to. For example:

GCF of 21 and 28 is 7 (since 7 is the largest number that divides both 21 and 28).

GCF of 21 and 63 is 21 (since 21 is the largest number that divides both 21 and 63).

Hence, the GCF of 21 varies depending on the other number.

2.What are the common factors of 21 and 14? and How many common factors do 21 and 14 have?

List the factors of 21: 1, 3, 7, 21.

List the factors of 14: 1, 2, 7, 14.

Identify the common factors: 1 and 7.

Hence, the common factors of 21 and 14 are 1 and 7. Besides, 21 and 14 have 2 common factors.

3. What is the Greast Common Factor of 21 and 14?

List the factors of 14: 1, 2, 7, 14.

List the factors of 21: 1, 3, 7, 21.

Identify the common factors: 1 and 7.

The greatest common factor (GCF) is the largest common factor, which is 7.

Hence, the GCF of 14 and 21 is 7.

4. A teacher has 21 students and wants to divide them into equal groups for a project. If each group must have the same number of students, how many students can be in each group? List all the possible ways.

Use the factors of 21: 1, 3, 7, 21.

List possible group sizes:

1. 1 group of 21 students.
2. 3 groups of 7 students each.
3. 7 groups of 3 students each.
4. 21 groups of 1 student each.

Hence, the students can be divided into groups of 1, 3, 7, or 21.

Common Mistakes and How to Avoid it

1. Forgetting 1 and the Number Itself

Example: A student lists the factors of 21 as 3 and 7, forgetting 1 and 21.

Why it’s wrong: Every number has at least two factors: 1 and itself.

How to avoid it: Always start by writing 1 and the number itself as factors. For 21, start with 1 and 21, then find the others.

2. Missing Factors

Example: A student thinks the factors of 21 are only 1, 7, and 21, missing 3.

Why it’s wrong: Factors come in pairs. If you find one factor, there’s always another that pairs with it.

How to avoid it: Use the multiplication or division method to check all possible pairs. For 21, test numbers like 3 (since 3 × 7 = 21).

3. Confusing Factors with Multiples

Example: A student says 42 is a factor of 21 because 21 × 2 = 42.

Why it’s wrong: Factors are smaller than or equal to the number, while multiples are larger.

How to avoid it: Remember that factors divide the number evenly, while multiples are the result of multiplying the number by another whole number.

4. Ignoring Negative Factors

Example: A student lists only positive factors of 21, forgetting negative pairs like (-3) × (-7) = 21.

Why it’s wrong: Factors can be negative if they multiply to give the original number.

How to avoid it: When asked for all factors, include both positive and negative pairs.

By being aware of these common mistakes and following the tips to avoid them, you’ll become more confident and accurate when working with factors. Keep practicing, and you’ll master this topic in no time!

Frequently Asked Questions:

1.What Is the Sum of Factors of 21?

The factors of 21 are 1, 3, 7, and 21. To find the sum of the factors, add them together: 1 + 3 + 7 + 21 = 32. Hence, the sum of the factors of 21 is 32.

2.What Is the Difference Between a Factor and a Multiple of 21?

Factor: A number that divides 21 evenly. For example, 3 is a factor of 21 because 21 ÷ 3 = 7.

Multiple: A number that is the result of multiplying 21 by another whole number. For example, 42 is a multiple of 21 because 21 × 2 = 42.

Hence, factors divide 21, while multiples are the results of multiplying 21 by other numbers.

3.Is 21 a Square Number or a Composite Number?

Square Number: A number that is the result of multiplying an integer by itself. For example, 16 is a square number because 4 × 4 = 16.

21 is not a square number because no integer multiplied by itself equals 21.

Composite Number: A number that has more than two factors (1 and itself).

21 is a composite number because it has four factors: 1, 3, 7, and 21.

Hence, 21 is not a square number but is a composite number.

4.What are the multiples of 21?

Multiples of a number are obtained by multiplying that number by integers (1, 2, 3, 4, …).

The multiples of 21 are:

• 21×1=2121×1=21
• 21×2=4221×2=42
• 21×3=6321×3=63
• 21×4=8421×4=84
• 21×5=10521×5=105
• And so on…

Hence, the multiples of 21 are 21, 42, 63, 84, 105, …

Conclusion

In this article, we explored the factors of 21, learning how to find them using division and multiplication methods. We also covered prime factorization, factor trees, and factor pairs, along with examples and tips to avoid common mistakes. Understanding factors is a key step in building a strong math foundation.

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