Question

Which of the following prime factorizations represents the greatest common factor of 162, 378, and 414?

2 · 3 3
2 · 3 4 · 7 · 23
2 · 3 2 · 7 · 23
2 · 3 2

Answer 1

To find the greatest common factor (GCF) of the numbers 162, 378, and 414, we first need to determine their prime factorizations.

1. **Factorization of 162**:
- \(162 = 2 \cdot 81\)
- \(81 = 3^4\)
- Therefore, \(162 = 2^1 \cdot 3^4\).

2. **Factorization of 378**:
- \(378 = 2 \cdot 189\)
- \(189 = 3 \cdot 63\)
- \(63 = 3 \cdot 21\)
- \(21 = 3 \cdot 7\)
- Therefore, \(378 = 2^1 \cdot 3^3 \cdot 7^1\).

3. **Factorization of 414**:
- \(414 = 2 \cdot 207\)
- \(207 = 3 \cdot 69\)
- \(69 = 3 \cdot 23\)
- Therefore, \(414 = 2^1 \cdot 3^2 \cdot 23^1\).

Now that we have the prime factorizations:
- \(162 = 2^1 \cdot 3^4\)
- \(378 = 2^1 \cdot 3^3 \cdot 7^1\)
- \(414 = 2^1 \cdot 3^2 \cdot 23^1\)

Next, we find the GCF by taking the lowest power of each prime that appears in all three factorizations:
- For the prime \(2\): \( \min(1, 1, 1) = 1\) → \(2^1\)
- For the prime \(3\): \( \min(4, 3, 2) = 2\) → \(3^2\)
- The primes \(7\) and \(23\) do not appear in all three numbers, so we do not include them.

Thus, the GCF is:
\[
GCF = 2^1 \cdot 3^2
\]

Combining these gives:
\[
GCF = 2 \cdot 3^2 = 2 \cdot 9 = 18
\]

In prime factorization form, this is represented as:
\[
2^1 \cdot 3^2
\]

Looking at the options provided:
- \(2 \cdot 3^2\)

is the correct representation for the greatest common factor.

So the correct choice is:
\[
\text{2} \cdot \text{3}^2
\]