The number of factors of a number `N=2^(3)xx3^(2)xx5^(3)` is

Options

A. 18
B. 45
C. 48
D. 9

Answer

To find the number of factors of the number \( N = 2^3 \times 3^2 \times 5^3 \), we can follow these steps:

### Step 1: Identify the prime factorization
The number \( N \) is already given in its prime factorization form:
\[ N = 2^3 \times 3^2 \times 5^3 \]

### Step 2: Use the formula for finding 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:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \]
where \( e_1, e_2, e_3 \) are the exponents of the prime factors.

### Step 3: Apply the formula
For our number:
- The exponent of 2 is \( 3 \) (i.e., \( e_1 = 3 \))
- The exponent of 3 is \( 2 \) (i.e., \( e_2 = 2 \))
- The exponent of 5 is \( 3 \) (i.e., \( e_3 = 3 \))

Now we can substitute these values into the formula:
\[ (3 + 1)(2 + 1)(3 + 1) \]

### Step 4: Calculate each term
Calculating each term:
- \( 3 + 1 = 4 \)
- \( 2 + 1 = 3 \)
- \( 3 + 1 = 4 \)

### Step 5: Multiply the results
Now, we multiply these results together:
\[ 4 \times 3 \times 4 \]

### Step 6: Perform the multiplication
Calculating this step by step:
1. First, calculate \( 4 \times 3 = 12 \)
2. Then, calculate \( 12 \times 4 = 48 \)

### Conclusion
Thus, the total number of factors of the number \( N = 2^3 \times 3^2 \times 5^3 \) is \( 48 \).

---