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

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.

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.

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:

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

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.

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:

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.

• 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?

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.