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, 72 and 99, using their prime factorization. The GCD is the largest number that divides both numbers exactly.

Step-by-Step Plan:

1. Prime Factorization: Break down both 72 and 99 into their prime factors.
2. Identify Common Prime Factors: Find the prime numbers that appear in both factorizations.
3. Find the GCD: Multiply the common prime factors, using the lowest power for each, to get the GCD.

Let's start with the first step.

Step 1: Prime Factorization

• 72 can be written as $72 = 2^3 \times 3^2$
• 99 can be written as $99 = 3^2 \times 11$

Let's confirm these factorizations. Amanita, the prime factorizations are correct:

• $72 = 2^3 \times 3^2$
• $99 = 3^2 \times 11$

Step 2: Identify Common Prime Factors

Let's look for the prime numbers that both numbers share. Both 72 and 99 have the prime number 3 in their factorizations. The lowest power of 3 in both is $3^2$.

Step 3: Find the GCD

The GCD is found by multiplying the common prime factors with their lowest powers:

$\text{GCD}(72, 99) = 3^2 = 9$

Let's confirm that 9 divides both 72 and 99 exactly. Amanita, both 72 and 99 can be divided exactly by 9:

• $72 \div 9 = 8$
• $99 \div 9 = 11$

So, the greatest common divisor (GCD) of 72 and 99 is:

$$\boxed{9}$$

This means 9 is the largest number that can divide both 72 and 99 without leaving a remainder!