To find the greatest common factor (GCF) of \( 45a^2b \) and \( 15ab \), we need to consider both the numerical coefficients and the algebraic variables.
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Find the GCF of the numerical coefficients:
- The coefficients are 45 and 15.
- The factors of 45 are \( 1, 3, 5, 9, 15, 45 \).
- The factors of 15 are \( 1, 3, 5, 15 \).
- The greatest common factor of 15 and 45 is 15.
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Find the GCF of the variables:
- For \( a^2 \) and \( a \), the lowest power of \( a \) is \( a^{\min(2, 1)} = a^1 = a \).
- For \( b \) and \( b \), the lowest power of \( b \) is \( b^{\min(1, 1)} = b^1 = b \).
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Combine the results:
- The GCF of the numerical part is 15.
- The GCF of the variable part is \( ab \).
Putting it all together, the GCF of \( 45a^2b \) and \( 15ab \) is: \[ 15ab \]
Thus, the greatest common factor is \( \boxed{15ab} \).