Find the number of factors of factors of the following :
7056

Answer

To find the number of factors of the number 7056, we will follow these steps:

### Step 1: Find the Prime Factorization of 7056
To find the prime factors of 7056, we can divide it by the smallest prime numbers until we reach 1.

1. **Divide by 2** (the smallest prime number):
- 7056 ÷ 2 = 3528
- 3528 ÷ 2 = 1764
- 1764 ÷ 2 = 882
- 882 ÷ 2 = 441 (no longer divisible by 2)

2. **Divide by 3** (next smallest prime number):
- 441 ÷ 3 = 147
- 147 ÷ 3 = 49 (no longer divisible by 3)

3. **Divide by 7**:
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1

Thus, the prime factorization of 7056 is:
\[ 7056 = 2^4 \times 3^2 \times 7^2 \]

### Step 2: Write the Prime Factorization in the Standard Form
From the factorization, we have:
- The prime factors are 2, 3, and 7.
- Their respective powers are 4, 2, and 2.

### Step 3: Use the Formula to Find the Total Number of Factors
The formula to find the total number of factors (n) from the prime factorization \( p_1^{a_1} \times p_2^{a_2} \times p_3^{a_3} \) is:
\[ n = (a_1 + 1)(a_2 + 1)(a_3 + 1) \]

In our case:
- For \( 2^4 \): \( a_1 = 4 \) → \( 4 + 1 = 5 \)
- For \( 3^2 \): \( a_2 = 2 \) → \( 2 + 1 = 3 \)
- For \( 7^2 \): \( a_3 = 2 \) → \( 2 + 1 = 3 \)

Now, substituting these values into the formula:
\[ n = (4 + 1)(2 + 1)(2 + 1) = 5 \times 3 \times 3 \]

### Step 4: Calculate the Total Number of Factors
Now we calculate:
\[ n = 5 \times 3 = 15 \]
\[ n = 15 \times 3 = 45 \]

Thus, the total number of factors of 7056 is **45**.

### Step 5: Find the Number of Factors of the Total Number of Factors
Now we need to find the number of factors of 45.

1. **Prime Factorization of 45**:
- 45 = 3^2 × 5^1

2. **Use the Formula Again**:
- For \( 3^2 \): \( a_1 = 2 \) → \( 2 + 1 = 3 \)
- For \( 5^1 \): \( a_2 = 1 \) → \( 1 + 1 = 2 \)

Now, substituting these values into the formula:
\[ n = (2 + 1)(1 + 1) = 3 \times 2 = 6 \]

Thus, the number of factors of the total number of factors of 7056 is **6**.

### Final Answer
The number of factors of the number 7056 is **6**.

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