Total number of 480 that are of the form 4n+2, `n ge 0`, is equal to

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem of finding the total number of divisors of 480 that are of the form \(4n + 2\) where \(n \geq 0\), we will follow these steps:

### Step 1: Factorize 480
First, we need to factorize the number 480 into its prime factors.

\[ 480 = 12 \times 40 = 12 \times (4 \times 10) = (3 \times 2^2) \times (2^2 \times 5) = 2^5 \times 3^1 \times 5^1 \]

### Step 2: Identify the form \(4n + 2\)
The expression \(4n + 2\) can be rewritten as:

\[ 4n + 2 = 2(2n + 1) \]

This indicates that any number of the form \(4n + 2\) is an even number that is not divisible by 4 (since \(2n + 1\) is an odd integer).

### Step 3: Determine the divisors of 480 that fit the form
To find the divisors of 480 that are of the form \(4n + 2\), we need to consider the divisors of 480 that are even but not divisible by 4.

### Step 4: Count the even divisors
The even divisors of 480 can be expressed in terms of its prime factorization. An even divisor must include at least one factor of 2. The general form of an even divisor of 480 is:

\[ 2^a \times 3^b \times 5^c \]

where:
- \(1 \leq a \leq 5\) (since it must be even),
- \(0 \leq b \leq 1\),
- \(0 \leq c \leq 1\).

### Step 5: Exclude divisors that are multiples of 4
To ensure that the divisor is not a multiple of 4, \(a\) can only be 1 (since \(a = 2\) or more would make it a multiple of 4). Thus, we have:

- \(a = 1\) (only one choice),
- \(b\) can be 0 or 1 (2 choices),
- \(c\) can be 0 or 1 (2 choices).

### Step 6: Calculate the total number of suitable divisors
The total number of divisors of the form \(4n + 2\) is given by multiplying the number of choices for \(b\) and \(c\):

\[ \text{Total} = 1 \times 2 \times 2 = 4 \]

### Final Answer
Thus, the total number of divisors of 480 that are of the form \(4n + 2\) is **4**.

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