Find the number of positive integral values of x satisfying `[x/9]=[x/11]` is where [.] -=Gl.F) (a). 21 (b). 22 (c). 23 (d). 24

Options

A. 21
B. 22
C. 23
D. 24

Answer

To solve the problem of finding the number of positive integral values of \( x \) satisfying the equation \( \left\lfloor \frac{x}{9} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor \), we can follow these steps:

### Step 1: Understand the Greatest Integer Function
The greatest integer function \( \left\lfloor y \right\rfloor \) gives the largest integer less than or equal to \( y \). Therefore, we need to find intervals where \( \left\lfloor \frac{x}{9} \right\rfloor \) and \( \left\lfloor \frac{x}{11} \right\rfloor \) are equal.

### Step 2: Set Up the Equation
Let \( n = \left\lfloor \frac{x}{9} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor \). This means:
\[ n \leq \frac{x}{9} < n+1 \quad \text{and} \quad n \leq \frac{x}{11} < n+1 \]

### Step 3: Express \( x \) in Terms of \( n \)
From the inequalities, we can express \( x \):
1. From \( n \leq \frac{x}{9} < n+1 \):
\[ 9n \leq x < 9(n+1) \]
2. From \( n \leq \frac{x}{11} < n+1 \):
\[ 11n \leq x < 11(n+1) \]

### Step 4: Find the Intersection of the Intervals
We need to find the intersection of the two intervals:
\[ [9n, 9(n+1)) \quad \text{and} \quad [11n, 11(n+1)) \]
The intersection will give us the values of \( x \) that satisfy both conditions.

### Step 5: Determine the Range of \( x \)
The intersection can be expressed as:
\[ \max(9n, 11n) \leq x < \min(9(n+1), 11(n+1)) \]
This simplifies to:
\[ 11n \leq x < 9n + 9 \]
Thus, the range of \( x \) is:
\[ 11n \leq x < 9n + 9 \]

### Step 6: Calculate the Length of the Interval
To find the number of integral values of \( x \) in this interval:
\[ \text{Length} = (9n + 9) - 11n = 9 - 2n \]
This length must be positive, so:
\[ 9 - 2n > 0 \implies n < 4.5 \]
Thus, \( n \) can take values \( 0, 1, 2, 3, 4 \).

### Step 7: Count the Values for Each \( n \)
1. For \( n = 0 \): \( 0 \leq x < 9 \) → 9 values (1 to 8)
2. For \( n = 1 \): \( 11 \leq x < 18 \) → 7 values (11 to 17)
3. For \( n = 2 \): \( 22 \leq x < 27 \) → 5 values (22 to 26)
4. For \( n = 3 \): \( 33 \leq x < 36 \) → 3 values (33 to 35)
5. For \( n = 4 \): \( 44 \leq x < 49 \) → 1 value (44)

### Step 8: Total the Values
Now, we sum the values:
\[ 8 + 7 + 5 + 3 + 1 = 24 \]

### Final Answer
Thus, the total number of positive integral values of \( x \) satisfying the equation is \( \boxed{24} \).