Question

gcf of 12ab^3 and 90a and 150b and 162ab

Answer 1

To find the greatest common factor (GCF) of the expressions \(12ab^3\), \(90a\), \(150b\), and \(162ab\), we will first factor each term.

1. **Factor each coefficient**:
- \(12 = 2^2 \times 3\)
- \(90 = 2 \times 3^2 \times 5\)
- \(150 = 2 \times 3 \times 5^2\)
- \(162 = 2 \times 3^4\)

2. **Combine the variable factors**:
- \(12ab^3\) has factors \(a^1b^3\)
- \(90a\) has factors \(a^1\)
- \(150b\) has factors \(b^1\)
- \(162ab\) has factors \(a^1b^1\)

3. **Finding the GCF of the coefficients**:
- The minimum power of \(2\) from the terms is \(2^0\) (since \(90\) and \(150\) don’t contribute a factor of \(2\)).
- The minimum power of \(3\) is \(3^1\), which appears in each term.
- The factor of \(5\) contributes \(5^0\) (since \(12ab^3\) and \(162ab\) do not have it).

Therefore, the GCF of the coefficients is:
\[
GCF = 2^0 \times 3^1 \times 5^0 = 3
\]

4. **Finding the GCF of the variables**:
- The minimum power of \(a\) in all terms is \(a^0\) (because of \(150b\)).
- The minimum power of \(b\) is \(b^0\) from \(90a\).

Therefore, the GCF of the variables is:
\[
a^0b^0 = 1
\]

5. **Combine the GCF of the coefficients and variables**:
The GCF of the entire expressions will be:
\[
GCF = 3 \cdot 1 = 3
\]

Thus, the greatest common factor (GCF) of \(12ab^3\), \(90a\), \(150b\), and \(162ab\) is \(\boxed{3}\).