What is the greatest common factor of 81 and 180?

The greatest common factor (GCF) of 81 and 180 is 9.

To find the GCF, we first need to determine the prime factorization of each number:

• 81: The prime factorization of 81 is 3 × 3 × 3 × 3, which can be written as 3⁴.
• 180: The prime factorization of 180 is 2 × 2 × 3 × 3 × 5, which can be written as 2² × 3² × 5.

Next, we identify the common prime factors and their lowest powers:

• The only common prime factor is 3.
• In 81, the power of 3 is 4 (3⁴), and in 180, the power of 3 is 2 (3²).

We take the lowest power of the common prime factor:

• For 3, the lowest power is 2.

Thus, the GCF is:

• 3² = 9.

In conclusion, the greatest common factor of 81 and 180 is 9.