The number of common prime factors of 75, 60, 105 is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the number of common prime factors of the numbers 75, 60, and 105, we will follow these steps:

### Step 1: Find the prime factors of 75
To find the prime factors of 75, we can start by dividing it by the smallest prime number, which is 5.

- 75 ÷ 5 = 15
- 15 ÷ 5 = 3
- 3 is a prime number.

So, the prime factorization of 75 is:
\[ 75 = 5^2 \times 3^1 \]
The prime factors of 75 are 5 and 3.

### Step 2: Find the prime factors of 60
Next, we will find the prime factors of 60.

- 60 ÷ 2 = 30 (2 is the smallest prime number)
- 30 ÷ 2 = 15
- 15 ÷ 3 = 5
- 5 is a prime number.

So, the prime factorization of 60 is:
\[ 60 = 2^2 \times 3^1 \times 5^1 \]
The prime factors of 60 are 2, 3, and 5.

### Step 3: Find the prime factors of 105
Now, we will find the prime factors of 105.

- 105 ÷ 3 = 35 (3 is the smallest prime number that divides 105)
- 35 ÷ 5 = 7
- 7 is a prime number.

So, the prime factorization of 105 is:
\[ 105 = 3^1 \times 5^1 \times 7^1 \]
The prime factors of 105 are 3, 5, and 7.

### Step 4: Identify the common prime factors
Now, we will identify the common prime factors among the three numbers:

- Prime factors of 75: 5, 3
- Prime factors of 60: 2, 3, 5
- Prime factors of 105: 3, 5, 7

The common prime factors among 75, 60, and 105 are:
- 3
- 5

### Step 5: Count the number of common prime factors
We have found that the common prime factors are 3 and 5. Therefore, the number of common prime factors is:
\[ \text{Number of common prime factors} = 2 \]

### Final Answer
The number of common prime factors of 75, 60, and 105 is **2**.

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