Find the number of factors of factors of the following :
18522

Answer

To find the number of factors of the number 18522, we will follow these steps:

### Step 1: Prime Factorization
First, we need to perform the prime factorization of 18522. We can do this by dividing the number by the smallest prime numbers until we reach 1.

1. **Divide by 2** (the smallest prime number):
\[ 18522 \div 2 = 9261 \]

2. **Divide by 3**:
\[ 9261 \div 3 = 3087 \]
\[ 3087 \div 3 = 1029 \]
\[ 1029 \div 3 = 343 \]

3. **Divide by 7**:
\[ 343 \div 7 = 49 \]
\[ 49 \div 7 = 7 \]
\[ 7 \div 7 = 1 \]

So, the prime factorization of 18522 is:
\[ 18522 = 2^1 \times 3^3 \times 7^3 \]

### Step 2: Finding the Number of Factors
To find the total number of factors of a number from its prime factorization, we use the formula:
\[ (n_1 + 1)(n_2 + 1)(n_3 + 1) \ldots \]
where \(n_1, n_2, n_3, \ldots\) are the powers of the prime factors.

From our factorization:
- The power of 2 is 1.
- The power of 3 is 3.
- The power of 7 is 3.

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

### Step 3: Calculate the Total Number of Factors
Now we calculate:
\[ 2 \times 4 = 8 \]
\[ 8 \times 4 = 32 \]

Thus, the total number of factors of 18522 is **32**.

### Step 4: Finding the Number of Factors of Factors
Now, we need to find the number of factors of the number 32.

1. **Prime Factorization of 32**:
\[ 32 = 2^5 \]

2. **Finding the Number of Factors of 32**:
Using the same formula:
\[ (5 + 1) = 6 \]

Thus, the number of factors of the number 32 is **6**.

### Final Answer
The number of factors of the number 18522 is **32**, and the number of factors of 32 is **6**.

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