A number N when factorized can be written as `N=p_(1)^(4)xxp_(2)^(3)xxp_(3)^(7)`. Find the number of perfect squares which are factors of N. (The 3 prime numbers `p_(1),p_(2),p_(3)gt2`)

Options

A. 12
B. 24
C. 36
D. 6

Answer

To find the number of perfect square factors of the number \( N = p_1^4 \times p_2^3 \times p_3^7 \), we will follow these steps:

### Step 1: Identify the powers of prime factors
The prime factorization of \( N \) is given as:
- \( p_1 \) has a power of 4
- \( p_2 \) has a power of 3
- \( p_3 \) has a power of 7

### Step 2: Determine the possible powers for perfect squares
A perfect square must have even powers for all its prime factors. Therefore, we need to consider the even powers that can be chosen from each of the prime factors.

- For \( p_1^4 \): The possible even powers are \( 0, 2, 4 \). This gives us 3 options.
- For \( p_2^3 \): The possible even powers are \( 0, 2 \). This gives us 2 options.
- For \( p_3^7 \): The possible even powers are \( 0, 2, 4, 6 \). This gives us 4 options.

### Step 3: Calculate the total number of combinations
To find the total number of perfect square factors, we multiply the number of choices for each prime factor:

\[ \text{Total perfect square factors} = (\text{choices for } p_1) \times (\text{choices for } p_2) \times (\text{choices for } p_3) \]

Substituting the values we found:

\[ \text{Total perfect square factors} = 3 \times 2 \times 4 \]

### Step 4: Perform the multiplication
Calculating the above expression:

\[ 3 \times 2 = 6 \]
\[ 6 \times 4 = 24 \]

### Conclusion
The total number of perfect square factors of \( N \) is **24**.

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