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# What are all the factors, the prime factorization, and factor pairs of 28?

To find the factors of 28, divide 28 by each number starting with 1 and working up to 28

Now let us find how to calculate all the factors of 28:
2
= 14
4
= 7
As you can see, the factors of 28 are:
1, 2, 4, 7, 14, 28

## Introduction to Factors

Factors are fundamental to the field of mathematics, particularly when it comes to understanding the nuances of numbers. Starting with the number 28 as an example, factors are the integers that, when multiplied together, produce the number 28. Delving into the study of these factors enlightens us on the constituent parts and inherent properties of 28 within the mathematical universe.

## What are Factors of 28?

The concept of factors pertains to numbers that divide 28 without leaving a remainder. Each factor of 28 pairs with another to reconstruct the original number when multiplied. For clarity and convenience, we provide a list of factors for 28:

• 1 × 28 = 28
• 2 × 14 = 28
• 4 × 7 = 28

Accordingly, the complete set of factors for 28 includes the integers: 1, 2, 4, 7, 14, and 28.

## How to Find the Factors of 28?

Identifying the factors of 28 can be achieved through the division method. Factors are integers which, after dividing 28, leave no remainder. The division should be done using integers up to and including 28 itself. Here’s how we can perform this operation:

• 28 ÷ 1 = 28
• 28 ÷ 2 = 14
• 28 ÷ 4 = 7
• 28 ÷ 7 = 4
• 28 ÷ 14 = 2
• 28 ÷ 28 = 1

Dividing the number 28 by other integers, for instance, 3 or 9, would not yield a whole number and thus are not considered factors of 28.

## Pair Factors of 28

To comprehend pair factors, we look for two integers whose product is the number 28. Both positive and negative integers can form such pairs:

Positive Pair Factors Negative Pair Factors
1 × 28 -1 × -28
2 × 14 -2 × -14
4 × 7 -4 × -7

This table effectively illustrates that the numbers (1, 28), (2, 14), and (4, 7) as well as their negatives are pairs that factor into 28.

## Prime Factorization of 28

The prime factorization of 28 distills the number down to its essential prime number constituents. Here’s the visual breakdown of the number into its prime factors using a factor tree:

• 28 → 2 × 14
• 14 → 2 × 7

Since 7 is a prime number, it cannot be broken down further, leaving us with 28’s prime factors: 2 × 2 × 7 (or 22 × 7).

## Important Points to Remember

In summary, there are several important points to keep in mind about the number 28 and its factors:

• Total number of factors: 6
• Prime factors: 2 and 7
• Pair factors: (1, 28), (2, 14), and (4, 7)
• Sum of all factors: 1 + 2 + 4 + 7 + 14 + 28 = 56

One interesting fact about 28 is that it is a perfect number, meaning the sum of its factors (excluding itself) equals the number. This reveals a unique balance in its arithmetic composition.

## Exercises

• What pair of factors can be multiplied to get 28?
• Is 15 a factor of 28? Why or why not?
• What is the sum of all the factors of 28 excluding 28 itself?

## Solutions to Exercises

1. (1, 28), (2, 14), and (4, 7) are pairs that when multiplied result in 28.
2. No, 15 is not a factor of 28. When you divide 28 by 15, it leaves a remainder, so 15 does not divide 28 evenly.
3. The sum of all the factors of 28 excluding 28 itself is 1 + 2 + 4 + 7 + 14 = 28, which interestingly equals the number itself.

### Are all factors of 28 prime numbers?

No, not all factors of 28 are prime. While 2 and 7 are prime factors, the number also has composite factors such as 4 and 14.

### What makes 28 special in terms of its factors?

28 is special because it is a perfect number; the sum of its factors, excluding the number itself, equals 28.

### Can a factor of 28 also be a multiple of 28?

No, a multiple of 28 is a result of multiplying 28 by another integer, while a factor of 28 is a number that divides 28 without a remainder.