The number of divisions of 9600 , including 1 and 9600 are

Options

A. 50
B. 48
C. 58
D. 46

Answer

To find the number of divisors of the number 9600, we will follow these steps:

### Step 1: Prime Factorization of 9600
We start by performing the prime factorization of 9600. We can divide 9600 by 2 (the smallest prime number) repeatedly until we can no longer divide evenly by 2:

- \(9600 \div 2 = 4800\)
- \(4800 \div 2 = 2400\)
- \(2400 \div 2 = 1200\)
- \(1200 \div 2 = 600\)
- \(600 \div 2 = 300\)
- \(300 \div 2 = 150\)
- \(150 \div 2 = 75\)

Now, we can no longer divide by 2. Next, we try dividing by the next smallest prime number, which is 3:

- \(75 \div 3 = 25\)

Finally, we divide by 5:

- \(25 \div 5 = 5\)
- \(5 \div 5 = 1\)

Thus, the prime factorization of 9600 is:
\[ 9600 = 2^7 \times 3^1 \times 5^2 \]

### Step 2: Using the Divisor Formula
To find the number of divisors of a number based on its prime factorization, we use the formula:
\[ \text{If } n = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k}, \text{ then the number of divisors } d(n) = (e_1 + 1)(e_2 + 1) \ldots (e_k + 1) \]

In our case:
- For \(2^7\), \(e_1 = 7\)
- For \(3^1\), \(e_2 = 1\)
- For \(5^2\), \(e_3 = 2\)

Now we apply the formula:
\[ d(9600) = (7 + 1)(1 + 1)(2 + 1) \]

### Step 3: Calculate the Number of Divisors
Now, we calculate each term:
- \(7 + 1 = 8\)
- \(1 + 1 = 2\)
- \(2 + 1 = 3\)

Now multiply these results together:
\[ d(9600) = 8 \times 2 \times 3 \]

Calculating this gives:
\[ d(9600) = 48 \]

### Conclusion
Thus, the total number of divisors of 9600, including 1 and 9600 itself, is **48**.

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