To find the greatest common factor (GCF) of 420 and 660, we will follow these steps:
### Step 1: Find the Prime Factorization of Each Number
- **Prime Factorization of 420**:
- Start by dividing 420 by the smallest prime number, which is 2.
- \( 420 \div 2 = 210 \)
- \( 210 \div 2 = 105 \) (2 is used again)
- Now, divide 105 by the next smallest prime number, which is 3.
- \( 105 \div 3 = 35 \)
- Next, divide 35 by the next smallest prime number, which is 5.
- \( 35 \div 5 = 7 \)
- Finally, 7 is a prime number itself.
Therefore, the prime factorization of 420 is:
\[
420 = 2^2 \times 3^1 \times 5^1 \times 7^1
\]
- **Prime Factorization of 660**:
- Start by dividing 660 by 2.
- \( 660 \div 2 = 330 \)
- \( 330 \div 2 = 165 \) (2 is used again)
- Now, divide 165 by 3.
- \( 165 \div 3 = 55 \)
- Next, divide 55 by 5.
- \( 55 \div 5 = 11 \)
- Finally, 11 is a prime number itself.
Therefore, the prime factorization of 660 is:
\[
660 = 2^2 \times 3^1 \times 5^1 \times 11^1
\]
### Step 2: Identify the Common Prime Factors
- Now, we will compare the prime factorizations of both numbers:
- For 420: \( 2^2, 3^1, 5^1, 7^1 \)
- For 660: \( 2^2, 3^1, 5^1, 11^1 \)
The common prime factors are:
- \( 2^2 \)
- \( 3^1 \)
- \( 5^1 \)
### Step 3: Calculate the GCF
- To find the GCF, we take the lowest power of each common prime factor:
- For \( 2 \): \( 2^2 \)
- For \( 3 \): \( 3^1 \)
- For \( 5 \): \( 5^1 \)
Now, multiply these together:
\[
GCF = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5
\]
Calculating this step-by-step:
- \( 4 \times 3 = 12 \)
- \( 12 \times 5 = 60 \)
Thus, the greatest common factor (GCF) of 420 and 660 is:
\[
\text{GCF} = 60
\]
### Final Answer:
The greatest common factor of 420 and 660 is **60**.
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