Question

List all the factors of \(28 .\) Then list all the factor pairs for 28

Short answer

Factors: 1, 2, 4, 7, 14, 28. Factor pairs: (1, 28), (2, 14), (4, 7).

Step-by-step solution

Determine Factors

To find all factors of 28, divide it by integer values starting from 1 up to 28. A number is a factor if the division gives an integer result.

Check Divisibility and List Factors

Divide 28 by each integer starting from 1:- 28 ÷ 1 = 28 (factor) - 28 ÷ 2 = 14 (factor)- 28 ÷ 3 ≠ integer- 28 ÷ 4 = 7 (factor)- 28 ÷ 5 ≠ integer - 28 ÷ 6 ≠ integer - 28 ÷ 7 = 4 (factor) - 28 ÷ 14 = 2 (factor) - 28 ÷ 28 = 1 (factor).

List All Factors

From the divisions, the factors of 28 are: 1, 2, 4, 7, 14, and 28.

Determine Factor Pairs

Each factor pair multiplies to 28:- (1, 28) because 1 x 28 = 28.- (2, 14) because 2 x 14 = 28.- (4, 7) because 4 x 7 = 28.

List All Factor Pairs

The factor pairs of 28 are: (1, 28), (2, 14), and (4, 7).

Key concepts

Divisibility

To determine if a number is a factor of another, we use the concept of divisibility. A number is divisible by another if the result of the division is an integer (a whole number with no decimal). This is how we identify factors.
For example, to check if 2 is a factor of 28, we divide 28 by 2. Since 28 ÷ 2 = 14, and 14 is an integer, 2 is a factor of 28.
As another example, dividing 28 by 3 gives us approximately 9.33. Since 9.33 is not an integer, 3 is not a factor of 28.
This process helps us systematically find every factor of a number by checking the divisibility. Start from 1 and continue up to the number itself.

Integer Division

Integer division is crucial when finding factors. Here, you're dividing one integer by another to see if the division results in a whole number.
When we divided 28 by integers starting from 1 up to 28, it gave us insights about which numbers are factors. For example:

• 28 ÷ 1 = 28 (result is an integer, so 1 is a factor)
• 28 ÷ 2 = 14 (result is an integer, so 2 is a factor)
• 28 ÷ 4 = 7 (result is an integer, so 4 is a factor)
• And so on...

If the result is not an integer, it means the divisor is not a factor. This step-by-step division ensures that we don't miss any potential factors.

Factor Pairs

Factor pairs are two numbers that, when multiplied together, equal the original number. They come from the set of factors of that number.
In our example, we identified the factors of 28 as 1, 2, 4, 7, 14, and 28. To form the factor pairs, we match these numbers such that their product is 28. The process looks like this:

• (1, 28) because 1 x 28 = 28
• (2, 14) because 2 x 14 = 28
• (4, 7) because 4 x 7 = 28

These pairs help us understand the relationships between factors and how they combine to form the original number. Hence, the factor pairs of 28 are (1, 28), (2, 14), and (4, 7).