Number of divisors of the form `4n + 2, n ge 0` which can divide 240 is :

Options

A. 4
B. 8
C. 10
D. 3

Answer

To solve the problem of finding the number of divisors of the form \(4n + 2\) that can divide 240, we can follow these steps:

### Step 1: Find the prime factorization of 240.
The first step is to determine the prime factorization of 240.

\[ 240 = 2^4 \times 3^1 \times 5^1 \]

### Step 2: Identify the divisors of 240.
To find the divisors of 240, we can use the prime factorization. The total number of divisors can be calculated using the formula:

\[ \text{Total Divisors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

where \(e_1, e_2, e_3\) are the powers of the prime factors.

For 240:
- \(e_1 = 4\) (from \(2^4\))
- \(e_2 = 1\) (from \(3^1\))
- \(e_3 = 1\) (from \(5^1\))

Calculating the total number of divisors:

\[ (4 + 1)(1 + 1)(1 + 1) = 5 \times 2 \times 2 = 20 \]

### Step 3: Identify the divisors of the form \(4n + 2\).
A number of the form \(4n + 2\) can be expressed as \(2(2n + 1)\). This means that the divisor must be even (since it has a factor of 2) and when divided by 2, the result must be an odd number.

### Step 4: List the even divisors of 240.
The even divisors of 240 can be found by considering the divisors that include at least one factor of 2. The even divisors of 240 are:

- 2
- 4
- 6
- 8
- 10
- 12
- 15
- 20
- 24
- 30
- 40
- 60
- 80
- 120
- 240

### Step 5: Check which of these even divisors are of the form \(4n + 2\).
Now we will check each even divisor to see if it can be expressed in the form \(4n + 2\):

- \(2 = 4(0) + 2\) (valid)
- \(4 = 4(1) + 0\) (not valid)
- \(6 = 4(1) + 2\) (valid)
- \(8 = 4(2) + 0\) (not valid)
- \(10 = 4(2) + 2\) (valid)
- \(12 = 4(3) + 0\) (not valid)
- \(15\) (not even)
- \(20 = 4(5) + 0\) (not valid)
- \(24 = 4(6) + 0\) (not valid)
- \(30 = 4(7) + 2\) (valid)
- \(40 = 4(10) + 0\) (not valid)
- \(60 = 4(15) + 0\) (not valid)
- \(80 = 4(20) + 0\) (not valid)
- \(120 = 4(30) + 0\) (not valid)
- \(240 = 4(60) + 0\) (not valid)

### Step 6: Count the valid divisors.
The valid divisors of the form \(4n + 2\) are:

- 2
- 6
- 10
- 30

Thus, there are **4 divisors** of the form \(4n + 2\) that can divide 240.

### Final Answer:
The number of divisors of the form \(4n + 2\) which can divide 240 is **4**.

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