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# What are all the factors, the prime factorization, and factor pairs of 21?

To find the factors of 21, divide 21 by each number starting with 1 and working up to 21

Now let us find how to calculate all the factors of 21:
3
= 7
As you can see, the factors of 21 are:
1, 3, 7, 21

## Introduction to Factors

Factors are the building blocks of numbers that, when multiplied together, result in a specific product. Taking 21 as our example, understanding the factors of 21 helps us to comprehend its structure and mathematical properties. Factors not only play a role in arithmetic but also in various applications such as solving equations and understanding number patterns.

## What are Factors of 21?

The concept of factors for 21 pertains to the whole numbers that can be evenly multiplied together to total 21. To illustrate this visually, one could arrange 21 objects into equal groupings, and the sizes of these possible groupings are the factors.

• 1 x 21 = 21
• 3 x 7 = 21

The complete list of factors for 21, therefore, is 1, 3, 7, and 21.

## How to Find the Factors of 21?

To find the factors of 21, one can employ the division method. This involves dividing 21 by integers less than or equal to it to seek out those which leave no remainder.

Here’s an example to demonstrate:

• 21 ÷ 1 = 21, thus 1 is a factor of 21.
• 21 ÷ 3 = 7, hence 3 is also a factor.
• 21 ÷ 7 = 3, which indicates that 7 is a factor too.
• 21 ÷ 21 = 1, and naturally, 21 is indeed a factor of itself.

Any other division would result in a fraction, which means no additional whole number factors exist for 21 beyond 1, 3, 7, and 21.

## Pair Factors of 21

Pair factors are two numbers that, when multiplied, result in the original number, which is 21 in our case. Below is a summary of both positive and negative pair factors:

Positive Pair Factors Negative Pair Factors
(1, 21) (-1, -21)
(3, 7) (-3, -7)

## Prime Factorization of 21

The prime factorization of a number involves breaking it down into its prime number constituents. For 21, the prime factorization process is as follows:

1. Divide 21 by the smallest prime number possible, which would be 3.
2. 21 ÷ 3 gives us 7, which is also a prime number.

This indicates that the prime factorization of 21 is 3 × 7.

A factor tree diagram would have 21 at the base and branch into 3 and 7 at the top, showcasing their role as prime factors.

## Important Points to Remember

When considering the factors of 21, remember these essential points:

• The total count of factors for 21 is four: 1, 3, 7, and 21.
• The prime factors of 21 are 3 and 7.
• Combined, the sum of all factors of 21 is 32.
• Understanding factors is crucial for various mathematical operations and number theory.

## Exercises

Test your knowledge with these exercises relating to the factors of 21:

• List all the factors of 21.
• Find two different pair factors for 21.
• What is the sum of all the factors of 21?

## Solutions to Exercises

Here are the solutions:

• The factors of 21 are 1, 3, 7, and 21.
• Pair factors of 21 could be (1, 21) and (3, 7).
• The sum of all factors of 21 is 1 + 3 + 7 + 21 = 32.