Find the number of factors of factors of the following :
7744

Answer

To find the number of factors of the number 7744, we will follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of 7744.

To do this, we can divide 7744 by the smallest prime number, which is 2, until we can no longer divide evenly:

- 7744 ÷ 2 = 3872
- 3872 ÷ 2 = 1936
- 1936 ÷ 2 = 968
- 968 ÷ 2 = 484
- 484 ÷ 2 = 242
- 242 ÷ 2 = 121
- 121 is not divisible by 2, so we try the next prime number, which is 11.
- 121 ÷ 11 = 11
- 11 ÷ 11 = 1

Thus, the prime factorization of 7744 is:
\[ 7744 = 2^6 \times 11^2 \]

### 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:
\[ (p_1 + 1)(p_2 + 1)(p_3 + 1)... \]
where \( p_1, p_2, p_3, \) etc. are the powers of the prime factors.

In our case:
- The power of 2 (which is \( p_1 \)) is 6.
- The power of 11 (which is \( p_2 \)) is 2.

Using the formula:
\[ \text{Total number of factors} = (6 + 1)(2 + 1) \]
\[ = 7 \times 3 \]
\[ = 21 \]

### Step 3: Finding the Number of Factors of the Factors
Now, we need to find the number of factors of 21 (the total number of factors we just calculated).

First, we find the prime factorization of 21:
\[ 21 = 3^1 \times 7^1 \]

Using the same formula for the number of factors:
- The power of 3 is 1.
- The power of 7 is 1.

So, we calculate:
\[ \text{Total number of factors of 21} = (1 + 1)(1 + 1) \]
\[ = 2 \times 2 \]
\[ = 4 \]

### Final Answer
The number of factors of the factors of 7744 is **4**.

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