Let's determine the greatest common factor (GCF) and the least common multiple (LCM) of the numbers \(A\), \(B\), and \(C\), given by:
[tex]\[
\begin{array}{l}
A = 2^8 \cdot 7^3 \cdot 17^7 \\
B = 3^7 \cdot 7 \cdot 13^7 \cdot 17^2 \cdot 19 \\
C = 3^5 \cdot 5^9 \cdot 7 \cdot 11^4 \cdot 19^8
\end{array}
\][/tex]
### Greatest Common Factor (GCF)
The GCF is found by taking the lowest power of each prime that appears in all three numbers.
1. Prime 2: Only appears in \(A\), so it does not contribute to the GCF.
2. Prime 3: Appears in \(B\) and \(C\), but not in \(A\), so it does not contribute.
3. Prime 5: Only appears in \(C\), so it does not contribute.
4. Prime 7: Appears in all three numbers. The lowest power is \(7^1\) (since \( B \text{ and } C \) both have \( 7^1 \)).
5. Prime 11: Only appears in \(C\), so it does not contribute.
6. Prime 13: Only appears in \(B\), so it does not contribute.
7. Prime 17: Appears in \(A\) and \(B\), but not in \(C\), so it does not contribute.
8. Prime 19: Appears in \(B\) and \(C\), but not in \(A\), so it does not contribute.
So, the GCF is:
[tex]\[
\operatorname{GCF}(A, B, C) = 7^1 = 7
\][/tex]
### Least Common Multiple (LCM)
The LCM is found by taking the highest power of each prime that appears in any of the numbers.
1. Prime 2: Highest power is \(2^8\) from \(A\).
2. Prime 3: Highest power is \(3^7\) from \(B\).
3. Prime 5: Highest power is \(5^9\) from \(C\).
4. Prime 7: Highest power is \(7^3\) from \(A\).
5. Prime 11: Highest power is \(11^4\) from \(C\).
6. Prime 13: Highest power is \(13^7\) from \(B\).
7. Prime 17: Highest power is \(17^7\) from \(A\).
8. Prime 19: Highest power is \(19^8\) from \(C\).
So, the LCM is:
[tex]\[
\operatorname{LCM}(A, B, C) = 2^8 \cdot 3^7 \cdot 5^9 \cdot 7^3 \cdot 11^4 \cdot 13^7 \cdot 17^7 \cdot 19^8
\][/tex]
### Final Answers:
[tex]\[
\begin{array}{l}
\operatorname{GCF}(A, B, C) = 7 \\
\operatorname{LCM}(A, B, C) = 2^8 \cdot 3^7 \cdot 5^9 \cdot 7^3 \cdot 11^4 \cdot 13^7 \cdot 17^7 \cdot 19^8
\end{array}
\][/tex]
