Identify the greatest common factor of 15, a15a and 35, a, b35ab.

Answer 1

To identify the greatest common factor (GCF) of the terms 15, \( a^{15} \), and 35, \( a \), \( b^{35}ab \), we need to examine the numerical and variable parts of each expression.

1. Numerical Components:

   • The numerical components are:
     • 15 = \( 3 \times 5 \)
     • 35 = \( 5 \times 7 \)
   • The GCF of the numerical parts:
     • The only common factor between 15 and 35 is 5.

2. Variable Components:

   • The variable components of the expressions are:

     • \( a^{15} \)
     • \( a \)
     • \( b^{35}ab = b^{35} \cdot a^1 \cdot b^1 = b^{36} \cdot a^1 \) (since \( b^{35}ab \) can be simplified to \( b^{35} \cdot a^1 \))

   • For the variable \( a \):

     • The minimum exponent of \( a \) from the terms is 1 (from \( a \)).

   • For the variable \( b \):

     • The term \( b^{36} \) does not appear in other factors, hence is not considered in the GCF.

3. Combining Numerical and Variable Components:

   • The GCF of the numerical components is 5, and the GCF for the variable \( a \) is \( a^1 \), while \( b \) does not contribute as discussed above.

Putting it all together, the greatest common factor of the given terms 15, \( a^{15} \), 35, \( a \), \( b^{35}ab \) is:

\[ \text{GCF} = 5a \]

So, the GCF is \( \boxed{5a} \).