Question

The greatest common factor of 42 and 36 is Write your answer in the space provided.

Short answer

The greatest common factor is 6.

Step-by-step solution

- List the Factors of Each Number

First, write down all the factors of each number. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

- Identify the Common Factors

Now, identify the factors that 42 and 36 have in common. These are the numbers that appear in both lists. The common factors are 1, 2, 3, and 6.

- Find the Greatest Common Factor

From the common factors identified in Step 2, select the largest one. The largest common factor is 6.

Key concepts

Factors

To grasp the concept of finding the greatest common factor (GCF), it's essential to first understand what factors are. Factors are whole numbers that are multiplied together to get another number. For example, the factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42 because these numbers can multiply in pairs to give the product of 42 (e.g., 1 x 42, 2 x 21, etc.).
To find these factors:

• Start with the number 1 and itself (since these are always factors).
• Proceed by testing each whole number to see if it divides evenly into the given number.
• Stop when you've crossed the halfway point (since the factors then begin to repeat).

Understanding this step makes it easier to identify common factors with other numbers.

Common Factors

Once you have the factors of each number, the next step is to identify the common factors. These are simply the numbers that appear in both sets of factors. Let's use 42 and 36 as examples: The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The common factors of 42 and 36 are the numbers that both lists share: 1, 2, 3, and 6.
Identifying common factors involves:

• Listing out the factors of each number.
• Comparing the lists to see which numbers appear in both.

This step helps us narrow down the factors to those that are relevant for finding the GCF.

Prime Factorization

Prime factorization can also be used to find the greatest common factor. Prime factorization is breaking down a number into its prime numbers. A prime number is a number greater than 1 that has no factors other than 1 and itself. For example, the prime factors of 42 and 36 are found as follows: 42 = 2 x 3 x 7, 36 = 2 x 2 x 3 x 3. Now, we look for the prime factors they have in common: both 42 and 36 have the factors 2 and 3.
The highest power of common prime factors gives us the GCF:

• For 2, the common power is 1 (since 42 has one 2 and 36 has two 2s, but we take the lower power).
• For 3, the common power is 1 (since both 42 and 36 have one 3).

Therefore, multiplying these together, the GCF is 2 x 3 = 6, which matches our earlier result using factor lists. Understanding how to use prime factorization is a powerful tool for solving GCF problems, especially with larger numbers.