How many positive factors of 76 are there?

Options

A. 6
B. 7
C. 5
D. 8

Answer

To find the number of positive factors of 76, we will follow these steps:

### Step 1: Prime Factorization of 76
We need to express 76 as a product of its prime factors.

- Start by dividing 76 by the smallest prime number, which is 2:
\[ 76 \div 2 = 38 \]
- Next, divide 38 by 2 again:
\[ 38 \div 2 = 19 \]
- Now, 19 is a prime number and cannot be divided further by any prime number other than itself.

So, we can express 76 as:
\[ 76 = 2^2 \times 19^1 \]

### Step 2: Using the Formula to Find the Number of Factors
To find the number of positive factors of a number, we use the formula based on its prime factorization. If a number \( n \) can be expressed as:
\[ n = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k} \]
where \( p_1, p_2, \ldots, p_k \) are the prime factors and \( e_1, e_2, \ldots, e_k \) are their respective powers, the number of positive factors \( d(n) \) is given by:
\[ d(n) = (e_1 + 1)(e_2 + 1) \ldots (e_k + 1) \]

### Step 3: Applying the Formula
From the prime factorization \( 76 = 2^2 \times 19^1 \):
- The exponent of 2 is 2, so \( e_1 = 2 \).
- The exponent of 19 is 1, so \( e_2 = 1 \).

Now, applying the formula:
\[ d(76) = (2 + 1)(1 + 1) = 3 \times 2 = 6 \]

### Conclusion
Therefore, the number of positive factors of 76 is **6**.

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