To find the greatest common divisor (GCD) of 205800 and 6468 using prime factorization:
1. **Prime factorize each number**: Write each number as a product of its prime factors.
For 205800: \( 205800 = 2^3 × 3^2 × 5^2 × 7^2 × 11 \)
For 6468: \( 6468 = 2^2 × 3 × 7 × 7 × 13 \)
2. **Identify common prime factors**: Find the prime factors that both numbers have in common.
In this case, the common prime factors are 2 and 3.
3. **Determine the lowest power of each common prime factor**: For each common prime factor, choose the smallest power it appears with in both factorizations.
For 2, it's \( 2^2 \).
For 3, it's \( 3^1 \).
4. **Multiply the lowest powers together**: Multiply the lowest powers of each common prime factor together to find the GCD.
\( GCD(205800, 6468) = 2^2 × 3^1 = 4 × 3 = 12 \)
So, the GCD of 205800 and 6468 is 12.
