Total number of values of `x,` of the form `1/n, n in N` in the interval `x in[1/25,1/10]` which satisfy the equation `{x} + {2x) + ......+ {12x}=78x` is `K.` then `K` is less than,(where { } represents fractional part function) (a)12 (b)13 (c)14 (d)15

Options

A. 12
B. 13
C. 14
D. 15

Answer

To solve the problem, we need to find the total number of values of \( x \) of the form \( \frac{1}{n} \) (where \( n \in \mathbb{N} \)) in the interval \( \left[\frac{1}{25}, \frac{1}{10}\right] \) that satisfy the equation:

\[ \{x\} + \{2x\} + \{3x\} + \ldots + \{12x\} = 78x \]

where \( \{.\} \) denotes the fractional part function.

### Step 1: Understanding the Equation
The fractional part \( \{kx\} \) can be expressed as:

\[ \{kx\} = kx - \lfloor kx \rfloor \]

Thus, we can rewrite the left-hand side of the equation as:

\[ \{x\} + \{2x\} + \ldots + \{12x\} = (x + 2x + 3x + \ldots + 12x) - (\lfloor x \rfloor + \lfloor 2x \rfloor + \ldots + \lfloor 12x \rfloor) \]

Calculating the sum of the first 12 terms gives:

\[ x + 2x + \ldots + 12x = \frac{12(12 + 1)}{2} x = 78x \]

So we have:

\[ \{x\} + \{2x\} + \ldots + \{12x\} = 78x - (\lfloor x \rfloor + \lfloor 2x \rfloor + \ldots + \lfloor 12x \rfloor) \]

### Step 2: Setting Up the Equation
From the equation, we can derive:

\[ \lfloor x \rfloor + \lfloor 2x \rfloor + \ldots + \lfloor 12x \rfloor = 0 \]

This implies that all terms \( \lfloor kx \rfloor \) must be zero, which means \( kx < 1 \) for all \( k = 1, 2, \ldots, 12 \).

### Step 3: Finding the Range of \( x \)
From \( kx < 1 \) for \( k = 1, 2, \ldots, 12 \):

- For \( k = 1 \): \( x < 1 \)
- For \( k = 2 \): \( x < \frac{1}{2} \)
- For \( k = 3 \): \( x < \frac{1}{3} \)
- ...
- For \( k = 12 \): \( x < \frac{1}{12} \)

The most restrictive condition is \( x < \frac{1}{12} \).

### Step 4: Analyzing the Interval
We are given the interval \( \left[\frac{1}{25}, \frac{1}{10}\right] \). We need to find values of \( x \) of the form \( \frac{1}{n} \) that lie within this interval and also satisfy \( x < \frac{1}{12} \).

### Step 5: Finding Valid Values of \( n \)
Now we check for values of \( n \):

1. \( \frac{1}{25} \) corresponds to \( n = 25 \)
2. \( \frac{1}{10} \) corresponds to \( n = 10 \)

We need \( n \) such that:

\[ 25 \leq n < 12 \]

The possible values of \( n \) are \( 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 \).

### Step 6: Counting Valid Values
The valid values of \( n \) are \( 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 \), which gives us a total of:

\[ 12 \text{ values} \]

### Conclusion
Thus, the total number of values of \( x \) is \( K = 12 \). Since the question asks for \( K \) being less than a certain number, the answer is:

**K is less than 13.**