How many factors of 420 are prime numbers?

Options

A. 3
B. 15
C. 7
D. 4

Answer

To find how many factors of 420 are prime numbers, we will follow these steps:

### Step 1: Prime Factorization of 420
We start by finding the prime factorization of 420.

1. Divide by the smallest prime number, which is 2:
\[ 420 \div 2 = 210 \]
2. Divide 210 by 2 again:
\[ 210 \div 2 = 105 \]
3. Now, divide 105 by the next smallest prime number, which is 3:
\[ 105 \div 3 = 35 \]
4. Finally, divide 35 by the next smallest prime number, which is 5:
\[ 35 \div 5 = 7 \]
5. Since 7 is a prime number, we stop here.

Thus, the prime factorization of 420 is:
\[ 420 = 2^2 \times 3^1 \times 5^1 \times 7^1 \]

### Step 2: Identify the Prime Factors
From the prime factorization, we can identify the prime factors of 420:
- 2
- 3
- 5
- 7

### Step 3: Count the Prime Factors
Now, we count the number of distinct prime factors:
- The prime factors are 2, 3, 5, and 7.

Thus, there are **4 distinct prime factors** of 420.

### Final Answer
The number of factors of 420 that are prime numbers is **4**.

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