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

The question asks us to find the greatest common divisor (GCD) of two numbers, 72 and 99, using their prime factorization. Here’s how we’ll solve it:

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 lowest powers of the common prime factors to get the GCD.

Let’s start with the first step.

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Step 1: Prime Factorization

Let’s find the prime factors of 72 and 99.

• 72 can be divided by 2 (the smallest prime number) repeatedly.
• 99 can be divided by 3 (the smallest prime number that divides it).

Let’s write their prime factorizations:

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

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Step 2: Identify Common Prime Factors

Now, let’s look for the prime numbers that both numbers share.

• Both 72 and 99 have the prime number 3 in their factorization.
• The lowest power of 3 in both is $3^2$.

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Step 3: Find the GCD

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

So, the GCD is:

$$3^2 = 9$$

Let’s confirm this by checking if 9 divides both 72 and 99.

• $72 \div 9 = 8$
• $99 \div 9 = 11$

Both are whole numbers, so 9 is indeed the GCD.

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Final Answer

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

$$\boxed{9}$$