What are the GCF and LCM of 30 and 24?

Answer

To find the GCF (Greatest Common Factor) and LCM (Least Common Multiple) of 30 and 24, we can follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of both numbers.

- **Prime factorization of 30**:
- 30 can be divided by 2: \(30 \div 2 = 15\)
- 15 can be divided by 3: \(15 \div 3 = 5\)
- 5 is a prime number.

Therefore, the prime factorization of 30 is:
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

- **Prime factorization of 24**:
- 24 can be divided by 2: \(24 \div 2 = 12\)
- 12 can be divided by 2: \(12 \div 2 = 6\)
- 6 can be divided by 2: \(6 \div 2 = 3\)
- 3 is a prime number.

Therefore, the prime factorization of 24 is:
\[ 24 = 2^3 \times 3^1 \]

### Step 2: Finding GCF
To find the GCF, we take the lowest power of all prime factors that appear in both factorizations.

- For \(2\): The minimum power is \(2^1\) (from 30).
- For \(3\): The minimum power is \(3^1\) (common to both).
- The factor \(5\) does not appear in 24, so we do not include it.

Thus, the GCF is:
\[ GCF = 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Step 3: Finding LCM
To find the LCM, we take the highest power of all prime factors that appear in either factorization.

- For \(2\): The maximum power is \(2^3\) (from 24).
- For \(3\): The maximum power is \(3^1\) (common to both).
- For \(5\): The maximum power is \(5^1\) (from 30).

Thus, the LCM is:
\[ LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \]
Calculating this:
\[ 8 \times 3 = 24 \]
\[ 24 \times 5 = 120 \]

### Final Answer
Therefore, the GCF of 30 and 24 is **6**, and the LCM is **120**.

---