When 200 is divided by a positive integer x, the remainder is 12. How many values of x are possible?

Options

A. 3
B. 8
C. 6
D. 2

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the problem
We are given that when 200 is divided by a positive integer \( x \), the remainder is 12. This means that we can express this relationship mathematically.

### Step 2: Set up the equation
Using the formula for division:
\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]
we can substitute the known values:
\[ 200 = x \times p + 12 \]
where \( p \) is the quotient.

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ x \times p = 200 - 12 \]
\[ x \times p = 188 \]

### Step 4: Factor 188
Next, we need to find the factors of 188 to determine the possible values of \( x \). We can start by performing the prime factorization of 188:
\[ 188 = 2 \times 94 \]
\[ 94 = 2 \times 47 \]
Thus, the prime factorization of 188 is:
\[ 188 = 2^2 \times 47 \]

### Step 5: List the factors of 188
The factors of 188 can be found by taking all combinations of the prime factors:
- \( 1 \)
- \( 2 \)
- \( 4 \)
- \( 47 \)
- \( 94 \)
- \( 188 \)

### Step 6: Apply the condition on the remainder
Since the remainder is 12, \( x \) must be greater than 12. Therefore, we only consider the factors of 188 that are greater than 12:
- \( 47 \)
- \( 94 \)
- \( 188 \)

### Step 7: Count the valid values of \( x \)
The valid values of \( x \) that satisfy the condition are \( 47 \), \( 94 \), and \( 188 \). Thus, there are a total of 3 possible values for \( x \).

### Final Answer
The total number of possible values of \( x \) is:
\[ \boxed{3} \]