The number of factors of 105 is:

Options

A. 8
B. 3
C. 4
D. 5

Answer

To find the number of factors of 105, we will follow these steps:

### Step 1: Prime Factorization of 105
First, we need to find the prime factorization of 105.

- 105 is not divisible by 2 (since it's odd).
- Check divisibility by 3: The sum of the digits (1 + 0 + 5 = 6) is divisible by 3. Thus, 105 is divisible by 3.

\[ 105 \div 3 = 35 \]

So, we have:
\[ 105 = 3 \times 35 \]

### Step 2: Factorization of 35
Next, we need to factor 35.

- 35 is not divisible by 2 (since it's odd).
- Check divisibility by 3: The sum of the digits (3 + 5 = 8) is not divisible by 3.
- Check divisibility by 5: 35 ends with a 5, so it is divisible by 5.

\[ 35 \div 5 = 7 \]

So, we have:
\[ 35 = 5 \times 7 \]

### Step 3: Complete Prime Factorization
Now we can write the complete prime factorization of 105:
\[ 105 = 3^1 \times 5^1 \times 7^1 \]

### Step 4: Use the Formula for Number of Factors
To find the number of factors, we use the formula:
\[ \text{Number of factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]
where \(e_1, e_2, e_3\) are the powers of the prime factors.

In our case:
- For \(3^1\), \(e_1 = 1\)
- For \(5^1\), \(e_2 = 1\)
- For \(7^1\), \(e_3 = 1\)

So, we calculate:
\[ (1 + 1)(1 + 1)(1 + 1) = 2 \times 2 \times 2 = 8 \]

### Conclusion
Therefore, the number of factors of 105 is **8**.

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