Gary applied the distributive property using the greatest common factor to determine the expression that is equivalent to 66 + 36. His work is shown below. Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66 Factors of 36: 1, 2,
3, 4, 6, 9, 12, 18, 36 66 + 36 = 3 (22 + 12
The Correct Answer and Explanation is:
Correct Answer:
Gary made a mistake. The correct expression using the greatest common factor (GCF) and the distributive property is:
66 + 36 = 6(11 + 6)
Explanation (300 words):
Gary attempted to apply the distributive property using the greatest common factor (GCF) to factor the expression 66 + 36. The distributive property states that for any numbers a, b, and c:
a × (b + c) = a × b + a × c
This can be used in reverse to factor expressions by identifying a common factor. In this case, we want to express 66 + 36 as:
GCF × (some expression)
First, identify the common factors of 66 and 36.
- Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
From this list, the greatest common factor of 66 and 36 is 6.
Now divide each number by the GCF:
- 66 ÷ 6 = 11
- 36 ÷ 6 = 6
So we rewrite the expression using the distributive property:
66 + 36 = 6 × (11 + 6)
66 + 36 = 6(17)
Gary incorrectly chose 3 as the common factor and wrote:
66 + 36 = 3(22 + 12)
While it is mathematically true that 3(22 + 12) = 66 + 36 = 102, it does not use the greatest common factor. The goal was to find the GCF, which is 6, and factor it out.
Using the GCF makes the resulting expression more simplified and useful for factoring or mental math. This method is often used in algebra to factor polynomials or simplify numeric expressions.
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