Problem 23 - Elementary Algebra

Question

For the following problems, specify all the whole number factors of each number. For example, the complete set of whole number factors of 6 is 1,2,3,6 . 21

Step-by-Step Solution

Answer

Answer: The whole number factors of 21 are 1, 3, 7, and 21.

Step 1: Understand the problem

We need to find all the whole number factors of the number 21. Factors are the numbers that can be multiplied together to get the original number, 21 in this case.

Step 2: Start with the lowest factors

The lowest factor of any number will always be 1. So, our first pair of factors is (1, 21), as 1*21 = 21.

Step 3: Continue finding factors

We will now continue to find the whole number factors by checking if the number is divisible by increasing numbers. Let's try with 2, the next whole number after 1. 21 is not divisible by 2 without a remainder, so 2 is not a factor of 21.

Step 4: Check for divisibility by the next number

Now we will try to divide 21 by 3. Since 21 is divisible by 3 without a remainder (21/3 = 7), our next pair of factors is (3, 7), as 3*7 = 21.

Step 5: Continue checking for divisibility

We will keep checking for divisibility by increasing numbers until we reach 7. As we've already found 7 to be a factor, there's no need to go beyond that. Any higher numbers would only result in repeating the same factor pairs.

Step 6: List the whole number factors

Based on the pairs of factors we found, the whole number factors of 21 are: 1, 3, 7, and 21. So, the complete set of whole number factors of 21 is 1, 3, 7, and 21.

Key Concepts

Understanding Divisibility

Divisibility is a key concept when finding the factors of a number. It refers to a scenario where one number can be divided by another without leaving a remainder. In other words, a number is divisible by another if, after division, the result is a whole number.

In the case of 21, we need to identify all numbers it is divisible by. We begin with the smallest possible whole number, which is 1. Since all numbers are divisible by 1, 21 divided by 1 yields 21, confirming that both 1 and 21 are factors. This understanding is crucial as it lays the foundation for identifying other factors.

• Start with the smallest numbers and work your way up.
• Ensure the division results in a whole number.
• Stop once you reach factors you've already identified (in this case, past 7).

Understanding and applying divisibility rules helps streamline this process, making it easier to determine the factors.

Multiplication and Factor Pairs

Multiplication and factor pairs go hand in hand. A factor pair consists of two numbers that, when multiplied together, equal the original number. To find these pairs, think about numbers that go into the original number without any leftovers.

When identifying factors for 21, you first check the basic pair: 1 and 21. Multiplying these gives the original number back, naturally. Next, by testing other small numbers like 3 and determining that it can multiply with 7 to make 21, we find another factor pair. These multiplication checks confirm the factor pairs, allowing us to say that \(3 \times 7 = 21\) is valid, just like \(1 \times 21 = 21\) is.

• Identify initial factor pairs starting from the lowest numbers (e.g., 1).
• Multiply potential factors to verify they result in the original number.
• Acknowledge pairs naturally repeat past a certain point (e.g., beyond 7 for 21).

So, the multiplication approach is about breaking down the number into smaller, manageable parts using pairs.

Identifying Pairs of Factors

Identifying pairs of factors involves finding all unique combinations of numbers that, when multiplied, give the original number. The concept emphasizes understanding how numbers relate through multiplication, essentially reversing the multiplication process.

For the number 21, we start with obvious pairs like 1 and 21. We then identify that dividing 21 by another whole number, such as 3, results in 7. This means both 3 and 7 are factors as well, creating the factor pair (3, 7). Once you reach 7, as we find in the solution, any further checks will simply reiterate known pairs, reinforcing the factors already determined.

• Discover the basic factor pair (e.g., 1 and the number itself).
• Check sequentially with numbers greater than 1 for additional pairs.
• Recognize that all factors appear as pairs in multiplication solutions.

By methodically testing and pairing, we can confidently list the factors of a number as the product of each factor pair.