If `0 lt x lt 1000 and [x/2]+[x/3]+[x/5]=31/30x`, (where `[.]` denotes the greatest integer function then number of possible values of x.

Options

A. 32
B. 33
C. 34
D. none of these

Answer

To solve the equation \(\left[\frac{x}{2}\right] + \left[\frac{x}{3}\right] + \left[\frac{x}{5}\right] = \frac{31}{30}x\) under the constraints \(0 < x < 1000\), we will follow these steps:

### Step 1: Understanding the Greatest Integer Function
The greatest integer function \([\cdot]\) gives the largest integer less than or equal to the argument. Thus, we can express:
- \(\left[\frac{x}{2}\right] = \frac{x}{2} - \{\frac{x}{2}\}\)
- \(\left[\frac{x}{3}\right] = \frac{x}{3} - \{\frac{x}{3}\}\)
- \(\left[\frac{x}{5}\right] = \frac{x}{5} - \{\frac{x}{5}\}\)

Where \(\{y\}\) denotes the fractional part of \(y\).

### Step 2: Rewrite the Equation
Substituting the expressions from Step 1 into the equation gives:
\[ \left(\frac{x}{2} - \{\frac{x}{2}\}\right) + \left(\frac{x}{3} - \{\frac{x}{3}\}\right) + \left(\frac{x}{5} - \{\frac{x}{5}\}\right) = \frac{31}{30}x \]
This simplifies to:
\[ \frac{x}{2} + \frac{x}{3} + \frac{x}{5} - \left(\{\frac{x}{2}\} + \{\frac{x}{3}\} + \{\frac{x}{5}\}\right) = \frac{31}{30}x \]

### Step 3: Calculate the Left Side
To combine the fractions on the left side, find a common denominator:
\[ \text{LCM}(2, 3, 5) = 30 \]
Thus:
\[ \frac{15x}{30} + \frac{10x}{30} + \frac{6x}{30} = \frac{31x}{30} \]
So we have:
\[ \frac{31x}{30} - \left(\{\frac{x}{2}\} + \{\frac{x}{3}\} + \{\frac{x}{5}\}\right) = \frac{31}{30}x \]

### Step 4: Simplifying the Equation
From the above equation, we can deduce:
\[ -\left(\{\frac{x}{2}\} + \{\frac{x}{3}\} + \{\frac{x}{5}\}\right) = 0 \]
This implies:
\[ \{\frac{x}{2}\} + \{\frac{x}{3}\} + \{\frac{x}{5}\} = 0 \]
Since the fractional parts are non-negative, this means:
\[ \{\frac{x}{2}\} = 0, \quad \{\frac{x}{3}\} = 0, \quad \{\frac{x}{5}\} = 0 \]

### Step 5: Finding Values of \(x\)
The conditions imply that \(x\) must be a multiple of \(2\), \(3\), and \(5\). The least common multiple of these numbers is:
\[ \text{LCM}(2, 3, 5) = 30 \]
Thus, \(x\) can be expressed as:
\[ x = 30n \quad \text{for integer } n \]

### Step 6: Determine the Range of \(n\)
Given the constraint \(0 < x < 1000\):
\[ 0 < 30n < 1000 \]
Dividing through by \(30\):
\[ 0 < n < \frac{1000}{30} \approx 33.33 \]
Thus, \(n\) can take integer values from \(1\) to \(33\).

### Step 7: Conclusion
The possible values of \(n\) are \(1, 2, 3, \ldots, 33\), giving us a total of \(33\) possible values for \(x\).

### Final Answer
The number of possible values of \(x\) is **33**.

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