How many positive integer values of 'a' are possible such that `(a+220)/(a+4)` is an integer?

Options

A. A) `16`
B. B) `6`
C. C) `12`
D. D) `18`

Answer

To solve the problem of how many positive integer values of 'a' are possible such that \((a + 220)/(a + 4)\) is an integer, we can follow these steps:

### Step 1: Set Up the Equation
We start with the expression:
\[ \frac{a + 220}{a + 4} \]
We want this to be an integer, which means that \(a + 4\) must divide \(a + 220\).

### Step 2: Rewrite the Expression
We can rewrite the expression as follows:
\[ \frac{a + 220}{a + 4} = 1 + \frac{216}{a + 4} \]
This shows that for the entire expression to be an integer, \(\frac{216}{a + 4}\) must also be an integer. This implies that \(a + 4\) must be a divisor of 216.

### Step 3: Find the Divisors of 216
Next, we need to find the divisors of 216. The prime factorization of 216 is:
\[ 216 = 2^3 \times 3^3 \]
To find the total number of factors, we use the formula for the number of divisors, which is \((e_1 + 1)(e_2 + 1)\) for each prime factor \(p^{e_i}\):
\[ (3 + 1)(3 + 1) = 4 \times 4 = 16 \]
So, there are 16 divisors of 216.

### Step 4: List the Divisors
The divisors of 216 are:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216.

### Step 5: Apply the Condition for 'a'
Since \(a + 4\) must be a divisor of 216, we need to ensure that \(a + 4 > 4\) (since \(a\) must be a positive integer). This means we can only consider the divisors greater than 4:
- The valid divisors are: 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216.

### Step 6: Calculate Possible Values of 'a'
Now, we subtract 4 from each of these divisors to find the corresponding values of \(a\):
- For 6: \(a = 6 - 4 = 2\)
- For 8: \(a = 8 - 4 = 4\)
- For 9: \(a = 9 - 4 = 5\)
- For 12: \(a = 12 - 4 = 8\)
- For 18: \(a = 18 - 4 = 14\)
- For 24: \(a = 24 - 4 = 20\)
- For 27: \(a = 27 - 4 = 23\)
- For 36: \(a = 36 - 4 = 32\)
- For 54: \(a = 54 - 4 = 50\)
- For 72: \(a = 72 - 4 = 68\)
- For 108: \(a = 108 - 4 = 104\)
- For 216: \(a = 216 - 4 = 212\)

### Step 7: Count the Valid Values of 'a'
Counting the valid values of \(a\), we find there are 12 positive integer values of \(a\) that satisfy the condition.

### Final Answer
Thus, the number of positive integer values of \(a\) is:
\[ \boxed{12} \]