To find the greatest common factor (GCF) of the given numbers 81, 120, and 240, we need to determine the largest number that can divide all three numbers without leaving a remainder.

Step 1: Prime Factorization
We start by finding the prime factorization of each number.

Prime factorization of 81:
81 = 3 * 3 * 3 * 3 = 3^4

Prime factorization of 120:
120 = 2 * 2 * 2 * 3 * 5 = 2^3 * 3 * 5

Prime factorization of 240:
240 = 2 * 2 * 2 * 2 * 3 * 5 = 2^4 * 3 * 5

Step 2: Identifying Common Factors
Next, we identify the common factors among the prime factorizations.

The common factors of 81, 120, and 240 are the prime factors that appear in all three numbers. From the prime factorizations, we can see that the common factors are 2, 3, and 5.

Step 3: Determining the GCF
To find the GCF, we multiply the common factors together.

GCF = 2 * 3 * 5 = 30

Therefore, the GCF of 81, 120, and 240 is 30.



To find the least common multiple (LCM) of the given numbers 81, 120, and 240, we need to determine the smallest number that is divisible by all three numbers.

Step 1: Prime Factorization
We start by finding the prime factorization of each number.

Prime factorization of 81:
81 = 3 * 3 * 3 * 3 = 3^4

Prime factorization of 120:
120 = 2 * 2 * 2 * 3 * 5 = 2^3 * 3 * 5

Prime factorization of 240:
240 = 2 * 2 * 2 * 2 * 3 * 5 = 2^4 * 3 * 5

Step 2: Identifying Unique Factors
Next, we identify the unique factors among the prime factorizations.

The unique factors of 81, 120, and 240 are the prime factors that appear in any one of the numbers, considering the highest power of each prime factor.

The unique factors are:
2^4, 3^4, and 5.

Step 3: Determining the LCM
To find the LCM, we multiply the unique factors together.

LCM = 2^4 * 3^4 * 5 = 16 * 81 * 5 = 6480

Therefore, the LCM of 81, 120, and 240 is 6480.

In summary,
- The GCF of 81, 120, and 240 is 30.
- The LCM of 81, 120, and 240 is 6480.