To find the greatest common divisor (GCD) of two numbers, you need to identify their common prime factors and the lowest power of each prime factor that appears in both numbers.

Here are the two numbers:

\[
\text{Number 1} = 3^{13} \times 5^{17}
\]
\[
\text{Number 2} = 2^{12} \times 7^{21}
\]

Notice that the prime factors of Number 1 are 3 and 5, while the prime factors of Number 2 are 2 and 7. Since these numbers do not share any common prime factors, the GCD is simply:

\[
\boxed{1}
\]

This is because the GCD of two numbers with no common prime factors is always 1.

Would you like further details or another example?

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Here are 5 related questions to expand on this concept:

1. How can you calculate the GCD if there are shared prime factors between two numbers?
2. What is the LCM of \(3^{13} \times 5^{17}\) and \(2^{12} \times 7^{21}\)?
3. How does the Euclidean algorithm handle numbers without common prime factors?
4. What is the significance of GCD in number theory?
5. Can the GCD be used to simplify fractions involving large exponents?

**Tip:** If two numbers share no common prime factors, their GCD is always 1!

Explore the solution to finding the GCD of two large numbers, 3^13 * 5^17 and 2^12 * 7^21, using prime factorization. This example covers key concepts in number theory and demonstrates how to calculate the GCD when the numbers have no common prime factors. Suitable for students in Grades 9-12.

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