The number of integral values of n (where n>=2) such that the equation 2n{x} = 3x + 2[x] has exactly five solutions (where [.] denotes the greatest integer function and {x} denotes the fractional part of x) is

Options

A. 2
B. 3
C. 4
D. 0

Explanation

To solve the equation \( 2n \{x\} = 3x + 2[x] \) and find the number of integral values of \( n \) (where \( n \geq 2 \)) such that the equation has exactly five solutions, we can follow these steps:

### Step 1: Rewrite the equation
We know that any real number \( x \) can be expressed as:
\[ x = [x] + \{x\} \]
where \( [x] \) is the greatest integer less than or equal to \( x \) and \( \{x\} \) is the fractional part of \( x \). Substituting this into the equation gives:
\[ 2n \{x\} = 3([x] + \{x\}) + 2[x] \]
This simplifies to:
\[ 2n \{x\} = 3[x] + 3\{x\} + 2[x] \]
Combining like terms results in:
\[ 2n \{x\} = 5[x] + 3\{x\} \]

### Step 2: Isolate the fractional part
Rearranging the equation, we have:
\[ 2n \{x\} - 3\{x\} = 5[x] \]
Factoring out \( \{x\} \):
\[ \{x\}(2n - 3) = 5[x] \]
Thus, we can express \( \{x\} \) as:
\[ \{x\} = \frac{5[x]}{2n - 3} \]

### Step 3: Determine the range of the fractional part
Since \( \{x\} \) must lie between 0 and 1, we have:
\[ 0 \leq \frac{5[x]}{2n - 3} < 1 \]
This leads to two inequalities:
1. \( 5[x] \geq 0 \) (which is always true since \( [x] \) is non-negative)
2. \( 5[x] < 2n - 3 \)

### Step 4: Analyze the second inequality
From \( 5[x] < 2n - 3 \), we can derive:
\[ [x] < \frac{2n - 3}{5} \]
Let \( k = [x] \). Thus:
\[ k < \frac{2n - 3}{5} \]
This means \( k \) can take integral values from 0 up to \( \left\lfloor \frac{2n - 3}{5} \right\rfloor \).

### Step 5: Count the number of integral solutions
The number of integral values \( k \) can take is:
\[ \left\lfloor \frac{2n - 3}{5} \right\rfloor + 1 \]
For the equation to have exactly 5 solutions, we set:
\[ \left\lfloor \frac{2n - 3}{5} \right\rfloor + 1 = 5 \]
This simplifies to:
\[ \left\lfloor \frac{2n - 3}{5} \right\rfloor = 4 \]

### Step 6: Solve the inequality
The condition \( \left\lfloor \frac{2n - 3}{5} \right\rfloor = 4 \) implies:
\[ 4 \leq \frac{2n - 3} < 5 \]
This leads to two inequalities:
1. \( 2n - 3 \geq 20 \) or \( 2n \geq 23 \) or \( n \geq 11.5 \)
2. \( 2n - 3 < 25 \) or \( 2n < 28 \) or \( n < 14 \)

### Step 7: Determine the integral values of \( n \)
Combining these inequalities gives:
\[ 12 \leq n < 14 \]
Thus, the integral values of \( n \) are \( 12 \) and \( 13 \).

### Step 8: Count the valid values
The valid integral values of \( n \) are:
- \( n = 12 \)
- \( n = 13 \)

Therefore, there are **2 integral values** of \( n \) that satisfy the condition.

### Final Answer
The number of integral values of \( n \) such that the equation has exactly five solutions is **2**.