Find the greatest common factor

)Find the greatest common factor (GCF) of 32 and 28 (b)Use the GCF to factor 32 – 28. (a) GCF = (b) 32 – 28 = \Box \times (\Box – \Box)

The Correct Answer and Explanation is:

Here are the solutions to the problems presented in the image.

(a) GCF = 4
(b) 32 – 28 = 4 x (8 – 7)

Explanation

Part (a): Finding the Greatest Common Factor (GCF)

The greatest common factor, or GCF, is the largest number that can divide into two or more numbers without leaving a remainder. To find the GCF of 32 and 28, we first need to list all the factors for each number.

• Factors of 32: We look for pairs of numbers that multiply to give 32.
  • 1 x 32
  • 2 x 16
  • 4 x 8
    The complete list of factors for 32 is 1, 2, 4, 8, 16, and 32.
• Factors of 28: Next, we find the pairs of numbers that multiply to give 28.
  • 1 x 28
  • 2 x 14
  • 4 x 7
    The complete list of factors for 28 is 1, 2, 4, 7, 14, and 28.

Now, we compare the two lists of factors to find the ones they have in common. The common factors of 32 and 28 are 1, 2, and 4. The greatest of these common factors is 4. Therefore, the GCF of 32 and 28 is 4.

Part (b): Using the GCF to Factor 32 – 28

Factoring an expression using the GCF involves rewriting the expression as the product of the GCF and a new expression in parentheses. This process is an application of the distributive property in reverse.

1. Place the GCF outside the parentheses. We determined the GCF is 4, so our factored expression will start like this: 4 × (… – …)
2. Find the terms inside the parentheses. To find the new terms, we divide each of the original numbers by the GCF (which is 4).
   • Divide the first term: 32 ÷ 4 = 8.
   • Divide the second term: 28 ÷ 4 = 7.
3. Complete the expression. We place these results inside the parentheses. This gives us the final factored form: 4 × (8 – 7).

We can verify our answer by checking if the original expression and the factored expression are equal.

• Original expression: 32 – 28 = 4.
• Factored expression: 4 × (8 – 7) = 4 × 1 = 4.
  Since both expressions equal 4, our factorization is correct.

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