How many non-negative integral values of x satisfy the equation `[x/5]=[x/7]` (Here [x] denotes the greatest integer less than or equal to x.
(For example (3.4) = 3 and [-2.3] = - 3).

Answer

To solve the equation \(\left\lfloor \frac{x}{5} \right\rfloor = \left\lfloor \frac{x}{7} \right\rfloor\), we will analyze the intervals defined by the greatest integer function for both sides of the equation.

### Step 1: Define the intervals for \(\left\lfloor \frac{x}{5} \right\rfloor\)

The greatest integer function \(\left\lfloor \frac{x}{5} \right\rfloor\) takes on integer values based on the intervals of \(x\):
- For \(0 \leq x < 5\), \(\left\lfloor \frac{x}{5} \right\rfloor = 0\)
- For \(5 \leq x < 10\), \(\left\lfloor \frac{x}{5} \right\rfloor = 1\)
- For \(10 \leq x < 15\), \(\left\lfloor \frac{x}{5} \right\rfloor = 2\)
- For \(15 \leq x < 20\), \(\left\lfloor \frac{x}{5} \right\rfloor = 3\)
- For \(20 \leq x < 25\), \(\left\lfloor \frac{x}{5} \right\rfloor = 4\)
- For \(25 \leq x < 30\), \(\left\lfloor \frac{x}{5} \right\rfloor = 5\)
- And so on...

### Step 2: Define the intervals for \(\left\lfloor \frac{x}{7} \right\rfloor\)

Similarly, we analyze the intervals for \(\left\lfloor \frac{x}{7} \right\rfloor\):
- For \(0 \leq x < 7\), \(\left\lfloor \frac{x}{7} \right\rfloor = 0\)
- For \(7 \leq x < 14\), \(\left\lfloor \frac{x}{7} \right\rfloor = 1\)
- For \(14 \leq x < 21\), \(\left\lfloor \frac{x}{7} \right\rfloor = 2\)
- For \(21 \leq x < 28\), \(\left\lfloor \frac{x}{7} \right\rfloor = 3\)
- For \(28 \leq x < 35\), \(\left\lfloor \frac{x}{7} \right\rfloor = 4\)
- And so on...

### Step 3: Set the intervals equal to each other

Now we find the values of \(x\) for which these two functions are equal:

1. **For \(\left\lfloor \frac{x}{5} \right\rfloor = 0\)**:
- \(0 \leq x < 5\)
- \(0 \leq x < 7\) (from \(\left\lfloor \frac{x}{7} \right\rfloor = 0\))
- Valid \(x\): \(0, 1, 2, 3, 4\) (5 values)

2. **For \(\left\lfloor \frac{x}{5} \right\rfloor = 1\)**:
- \(5 \leq x < 10\)
- \(7 \leq x < 14\)
- Valid \(x\): \(7, 8, 9\) (3 values)

3. **For \(\left\lfloor \frac{x}{5} \right\rfloor = 2\)**:
- \(10 \leq x < 15\)
- \(14 \leq x < 21\)
- Valid \(x\): \(14\) (1 value)

4. **For \(\left\lfloor \frac{x}{5} \right\rfloor = 3\)**:
- \(15 \leq x < 20\)
- \(21 \leq x < 28\)
- No valid \(x\) (0 values)

5. **For \(\left\lfloor \frac{x}{5} \right\rfloor = 4\)**:
- \(20 \leq x < 25\)
- \(28 \leq x < 35\)
- No valid \(x\) (0 values)

6. **For \(\left\lfloor \frac{x}{5} \right\rfloor = 5\)**:
- \(25 \leq x < 30\)
- \(35 \leq x < 42\)
- No valid \(x\) (0 values)

### Step 4: Count the total valid values of \(x\)

Adding up all the valid values:
- From \(\left\lfloor \frac{x}{5} \right\rfloor = 0\): 5 values
- From \(\left\lfloor \frac{x}{5} \right\rfloor = 1\): 3 values
- From \(\left\lfloor \frac{x}{5} \right\rfloor = 2\): 1 value
- From \(\left\lfloor \frac{x}{5} \right\rfloor = 3\): 0 values
- From \(\left\lfloor \frac{x}{5} \right\rfloor = 4\): 0 values
- From \(\left\lfloor \frac{x}{5} \right\rfloor = 5\): 0 values

Total valid values of \(x = 5 + 3 + 1 + 0 + 0 + 0 = 9\).

### Final Answer
The total number of non-negative integral values of \(x\) that satisfy the equation is **9**.

---