Question

1. what is the greatest common factor?

\\(2 \cdot 2 \cdot 2 \cdot 2 \cdot 3\\)
\\(2 \cdot 2 \cdot 2 \cdot 3 \cdot 3\\)

(a) \\(2 \cdot 2 \cdot 2\\)
(b) \\(2 \cdot 2 \cdot 3\\)
(c) \\(2 \cdot 2 \cdot 2 \cdot 3\\)
(d) \\(2 \cdot 2 \cdot 2 \cdot 3 \cdot 3\\)

Explanation:

Identify the given prime factorizations

We are given the prime factorizations of two numbers:
First number: \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 3\)
Second number: \(2 \cdot 2 \cdot 2 \cdot 3 \cdot 3\)

Find the common prime factors

To find the greatest common factor (GCF), we look for the prime factors that appear in both factorizations.

• The factor \(2\) appears four times in the first number and three times in the second number. The common occurrence is three times: \(2 \cdot 2 \cdot 2\).
• The factor \(3\) appears once in the first number and twice in the second number. The common occurrence is once: \(3\).

Multiply the shared factors

We multiply the common prime factors together to get the greatest common factor:

$$ \text{GCF} = 2 \cdot 2 \cdot 2 \cdot 3 $$

Match with the given options

Comparing our result with the choices:

• (A) \(2 \cdot 2 \cdot 2\)
• (B) \(2 \cdot 2 \cdot 3\)
• (C) \(2 \cdot 2 \cdot 2 \cdot 3\)
• (D) \(2 \cdot 2 \cdot 2 \cdot 3 \cdot 3\)

The correct option is (C).

Answer:

• (A) \(2 \cdot 2 \cdot 2\)
• (B) \(2 \cdot 2 \cdot 3\)
• (C) \(2 \cdot 2 \cdot 2 \cdot 3\) (Correct answer)
• (D) \(2 \cdot 2 \cdot 2 \cdot 3 \cdot 3\)