Find the number of factors of factors of the following :
3875

Answer

To find the number of factors of the number 3875, we will follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of 3875.

- Start by dividing 3875 by the smallest prime number, which is 5:
\[ 3875 \div 5 = 775 \]
- Divide 775 by 5 again:
\[ 775 \div 5 = 155 \]
- Divide 155 by 5 once more:
\[ 155 \div 5 = 31 \]
- Now, 31 is a prime number.

So, the prime factorization of 3875 is:
\[ 3875 = 5^3 \times 31^1 \]

### Step 2: Finding the Number of Factors
To find the total number of factors, we use the formula:
\[ (n_1 + 1)(n_2 + 1) \ldots (n_k + 1) \]
where \( n_1, n_2, \ldots, n_k \) are the powers of the prime factors.

From our factorization:
- The power of 5 is 3.
- The power of 31 is 1.

Now, we apply the formula:
\[ (3 + 1)(1 + 1) = 4 \times 2 \]

### Step 3: Calculate the Total Number of Factors
Now, we calculate:
\[ 4 \times 2 = 8 \]

Thus, the total number of factors of 3875 is 8.

### Step 4: Finding the Number of Factors of the Factors
Now we need to find the number of factors of the number of factors we just calculated, which is 8.

First, we find the prime factorization of 8:
\[ 8 = 2^3 \]

Using the same formula for the number of factors:
- The power of 2 is 3.

So, we apply the formula again:
\[ (3 + 1) = 4 \]

Thus, the number of factors of the number of factors of 3875 is 4.

### Final Answer
The number of factors of the factors of 3875 is **4**.

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