Amanita, let's solve the problem step by step!

Understanding the Problem: The question asks us to find the greatest common divisor (GCD) of two numbers. The numbers are given in their prime factorized forms:

• $2^3 \times 5^2$
• $2^3 \times 3^2 \times 5$

Step-by-Step Plan:

1. Write out the prime factorization of both numbers.
2. Identify the common prime factors and their lowest powers.
3. Multiply these lowest powers together to get the GCD.
4. Calculate the value of the GCD.
5. Verify the answer.

Let's begin!

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Step 1: Write out the prime factorization

• The first number is $2^3 \times 5^2$.
• The second number is $2^3 \times 3^2 \times 5$.

Step 2: Identify the common prime factors and their lowest powers

• Both numbers have the prime factor $2$. The lowest power is $3$.
• Both numbers have the prime factor $5$. The lowest power is $1$.
• Only the second number has the prime factor $3$, so it is not included in the GCD.

So, the GCD is $2^3 \times 5^1$.

Step 3: Calculate the value of the GCD

Let's calculate $2^3 \times 5$. Amanita, the greatest common divisor (GCD) of $2^3 \times 5^2$ and $2^3 \times 3^2 \times 5$ is:

$$2^3 \times 5 = 8 \times 5 = 40$$

So, the answer is $40$. This means 40 is the largest number that can evenly divide both of the given numbers.