What is the greatest common divisor of $44100$ and $100800?$

To find the greatest common divisor (GCD) of $ 44100 $ and $ 100800 $, we can use the prime factorization method.

### Step 1: Prime Factorization of 44100
First, factorize $ 44100 $ into its prime factors:
\[
44100 = 2^2 \times 3^2 \times 5^2 \times 7^2
\]

### Step 2: Prime Factorization of 100800
Next, factorize $ 100800 $ into its prime factors:
\[
100800 = 2^6 \times 3^2 \times 5^2 \times 7
\]

### Step 3: Find the Common Prime Factors
The common prime factors between $ 44100 $ and $ 100800 $ are $ 2 $, $ 3 $, $ 5 $, and $ 7 $.

### Step 4: Determine the GCD
The GCD is found by taking the lowest power of each common prime factor:
\[
\text{GCD} = 2^2 \times 3^2 \times 5^2 \times 7 = 4 \times 9 \times 25 \times 7
\]

### Step 5: Calculate the Product
\[
\text{GCD} = 4 \times 9 = 36
\]
\[
36 \times 25 = 900
\]
\[
900 \times 7 = 6300
\]

So, the greatest common divisor of $ 44100 $ and $ 100800 $ is $ 6300 $.

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you find the least common multiple (LCM) of 44100 and 100800?
2. What is the GCD of three numbers: 44100, 100800, and 52920?
3. How does the Euclidean algorithm find the GCD of two numbers?
4. What is the role of prime factorization in finding the GCD?
5. Can the GCD of two numbers be greater than the smallest of the two numbers?

**Tip:** Understanding prime factorization is essential in solving problems related to GCD and LCM.

Learn how to find the greatest common divisor (GCD) of 44100 and 100800 using prime factorization method. Detailed step-by-step solution provided. Suitable for students in Grades 7-9.

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