Find the number of factors of factors of the following :
1008

Answer

To find the number of factors of the number 1008, we will follow these steps:

### Step 1: Prime Factorization of 1008
First, we need to find the prime factorization of 1008. We can do this by dividing 1008 by the smallest prime numbers until we reach 1.

- Divide by 2:
- 1008 ÷ 2 = 504
- 504 ÷ 2 = 252
- 252 ÷ 2 = 126
- 126 ÷ 2 = 63 (63 is not divisible by 2)

- Now divide by 3:
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7 (7 is a prime number)

So, the prime factorization of 1008 is:
\[ 1008 = 2^4 \times 3^2 \times 7^1 \]

### Step 2: Use the Formula to Find the Number of Factors
The formula to find the total number of factors of a number based on its prime factorization \( p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \) is:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

From our prime factorization:
- \( e_1 = 4 \) (for \( 2^4 \))
- \( e_2 = 2 \) (for \( 3^2 \))
- \( e_3 = 1 \) (for \( 7^1 \))

Now we can substitute these values into the formula:
\[ (4 + 1)(2 + 1)(1 + 1) \]

### Step 3: Calculate the Number of Factors
Now, we calculate:
- \( 4 + 1 = 5 \)
- \( 2 + 1 = 3 \)
- \( 1 + 1 = 2 \)

Now multiply these results:
\[ 5 \times 3 \times 2 = 30 \]

So, the total number of factors of 1008 is **30**.

### Step 4: Find the Number of Factors of the Number of Factors
Now, we need to find the number of factors of 30. First, we need to factor 30:
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

Using the same formula for the number of factors:
- \( e_1 = 1 \) (for \( 2^1 \))
- \( e_2 = 1 \) (for \( 3^1 \))
- \( e_3 = 1 \) (for \( 5^1 \))

Now substitute these values into the formula:
\[ (1 + 1)(1 + 1)(1 + 1) \]

Calculate:
- \( 1 + 1 = 2 \)

Now multiply:
\[ 2 \times 2 \times 2 = 8 \]

So, the total number of factors of 30 is **8**.

### Final Answer
The number of factors of the number of factors of 1008 is **8**.

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