To find the greatest common divisor (gcd) of each pair of integers $a$ and $b$, we need to look at the prime factorization of both $a$ and $b$ for each question and then take the minimum power of each common prime factor.

Problem 1:

Given:

• $a = 2^4 \cdot 3^4 \cdot 5^3$
• $b = 2^3 \cdot 3 \cdot 5^3$

The gcd will include each common prime factor raised to the lowest power in both factorizations:

• For $2$: $\min(4, 3) = 3$, so $2^3$
• For $3$: $\min(4, 1) = 1$, so $3^1$
• For $5$: $\min(3, 3) = 3$, so $5^3$

Thus, $\gcd(a, b) = 2^3 \cdot 3^1 \cdot 5^3 = 8 \cdot 3 \cdot 125 = 3000$

Problem 2:

Given:

• $a = 2^9 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13$
• $b = 5^9 \cdot 7^8 \cdot 13^3 \cdot 17$

The gcd will include each common prime factor raised to the lowest power in both factorizations:

• For $5$: $\min(1, 9) = 1$, so $5^1$
• For $7$: $\min(1, 8) = 1$, so $7^1$
• For $13$: $\min(1, 3) = 1$, so $13^1$

Thus, $\gcd(a, b) = 5^1 \cdot 7^1 \cdot 13^1 = 5 \cdot 7 \cdot 13 = 455$

Problem 3:

Given:

• $a = 2^4 \cdot 7$
• $b = 5^4 \cdot 13$

Here, there are no common prime factors between $a$ and $b$, so the gcd is simply $1$.

Thus, $\gcd(a, b) = 1$

Let me know if you need further clarification on any part of this solution or have additional questions!

Further Questions:

1. What is the method to find the gcd of numbers with more than two factors?
2. How does prime factorization help in finding the gcd?
3. Why do we take the minimum exponent for common factors when calculating gcd?
4. What would happen if one of the numbers has a factor that the other does not?
5. How can gcd calculations be applied in real-world problem-solving?

Tip:

When calculating gcd, always look for the lowest exponent of each common prime factor between two numbers.