The number of divisiors of 720 which are multiples of perfect square of first prime number is:

Options

A. 12
B. 14
C. 18
D. can't be determined

Answer

To find the number of divisors of 720 that are multiples of the perfect square of the first prime number (which is 2, and its square is 4), we can follow these steps:

### Step 1: Prime Factorization of 720
First, we need to factor 720 into its prime factors.

720 can be factored as follows:
- Start by dividing by 2:
\(720 \div 2 = 360\)
\(360 \div 2 = 180\)
\(180 \div 2 = 90\)
\(90 \div 2 = 45\)
- Now, 45 is not divisible by 2, so we divide by 3:
\(45 \div 3 = 15\)
\(15 \div 3 = 5\)
- Finally, we have 5, which is a prime number.

Thus, the prime factorization of 720 is:
\[ 720 = 2^4 \times 3^2 \times 5^1 \]

### Step 2: Understanding the Requirement
We need to find the divisors of 720 that are multiples of \(4\). Since \(4 = 2^2\), we are looking for divisors of the form \(2^a \times 3^b \times 5^c\) where \(a \geq 2\).

### Step 3: Determine the Range of Exponents
From the prime factorization:
- The exponent of 2 can be \(a = 2, 3, 4\) (since \(a\) must be at least 2 and can go up to 4).
- The exponent of 3 can be \(b = 0, 1, 2\) (since the maximum exponent of 3 is 2).
- The exponent of 5 can be \(c = 0, 1\) (since the maximum exponent of 5 is 1).

### Step 4: Count the Possible Combinations
Now we can count the number of choices for each exponent:
- For \(a\) (the exponent of 2), we have 3 choices: \(2, 3, 4\).
- For \(b\) (the exponent of 3), we have 3 choices: \(0, 1, 2\).
- For \(c\) (the exponent of 5), we have 2 choices: \(0, 1\).

### Step 5: Calculate the Total Number of Divisors
The total number of divisors that are multiples of \(4\) is given by multiplying the number of choices for each exponent:
\[ \text{Total divisors} = (\text{choices for } a) \times (\text{choices for } b) \times (\text{choices for } c) = 3 \times 3 \times 2 = 18 \]

Thus, the number of divisors of 720 that are multiples of \(4\) is **18**.

### Final Answer
The answer is **18**.

---