Identify the greatest common factor of 8, b8b and 14, a, b, c14abc.

Answer 1

To find the greatest common factor (GCF) of the numbers involved, we'll break down each term into its prime factors.

1. Identify the terms:

   • \(8\)
   • \(b8b\) (which we'll interpret as \(8b\) for our purposes)
   • \(14\)
   • \(a\)
   • \(b\)
   • \(c14abc\) (which we'll interpret as \(14abc\))

2. Factor each term:

   • \(8 = 2^3\)
   • \(8b = 2^3 \cdot b^1\)
   • \(14 = 2^1 \cdot 7^1\)
   • \(a = a^1\)
   • \(b = b^1\)
   • \(14abc = 2^1 \cdot 7^1 \cdot a^1 \cdot b^1 \cdot c^1\)

3. Identify the GCF:

   • To find the GCF, we look for the minimum power of each factor present in all terms.
   • For the factor \(2\):
     • The powers are \(3\) (in \(8\)), \(3\) (in \(8b\)), \(1\) (in \(14\)), and \(1\) (in \(14abc\)). The minimum power is \(1\).
   • For the factor \(7\):
     • The powers are \(0\) (in \(8\)), \(0\) (in \(8b\)), \(1\) (in \(14\)), and \(1\) (in \(14abc\)). The minimum power is \(0\).
   • For the factor \(b\):
     • The powers are \(0\) (in \(8\)), \(1\) (in \(8b\)), \(0\) (in \(14\)), and \(1\) (in \(14abc\)). The minimum power is \(0\).
   • For the factor \(a\):
     • The powers are \(0\) (in \(8\)), \(0\) (in \(8b\)), \(0\) (in \(14\)), and \(1\) (in \(14abc\)). The minimum power is \(0\).
   • For the factor \(c\):
     • The powers are \(0\) (in \(8\)), \(0\) (in \(8b\)), \(0\) (in \(14\)), and \(1\) (in \(14abc\)). The minimum power is \(0\).

4. Conclusion:

   • The GCF is \(2^1 = 2\).
   • Since there are no higher common factors from \(a\), \(b\), \(c\), or \(7\), these contribute \(1^0 = 1\).

Thus, the greatest common factor of \(8\), \(8b\), \(14\), \(a\), \(b\), and \(14abc\) is:

\[ \boxed{2} \]