If the number `98232x4567y` is divisible by `75`, then find the value of `(6x-5y)`.

Options

A. 21
B. 23
C. 25
D. 29

Answer

To solve the problem of finding the value of \(6x - 5y\) given that the number \(98232x4567y\) is divisible by \(75\), we need to check the conditions for divisibility by \(75\). The number \(75\) can be factored into \(25\) and \(3\), and we will check the divisibility rules for both.

### Step 1: Check divisibility by \(25\)
For a number to be divisible by \(25\), its last two digits must be \(00\), \(25\), \(50\), or \(75\). In our case, the last two digits of the number are \(7y\).

- If \(y = 0\), the last two digits are \(70\) (not divisible by \(25\)).
- If \(y = 1\), the last two digits are \(71\) (not divisible by \(25\)).
- If \(y = 2\), the last two digits are \(72\) (not divisible by \(25\)).
- If \(y = 3\), the last two digits are \(73\) (not divisible by \(25\)).
- If \(y = 4\), the last two digits are \(74\) (not divisible by \(25\)).
- If \(y = 5\), the last two digits are \(75\) (divisible by \(25\)).
- If \(y = 6\), the last two digits are \(76\) (not divisible by \(25\)).
- If \(y = 7\), the last two digits are \(77\) (not divisible by \(25\)).
- If \(y = 8\), the last two digits are \(78\) (not divisible by \(25\)).
- If \(y = 9\), the last two digits are \(79\) (not divisible by \(25\)).

Thus, the only possible value for \(y\) that makes \(98232x4567y\) divisible by \(25\) is \(y = 5\).

### Step 2: Check divisibility by \(3\)
For a number to be divisible by \(3\), the sum of its digits must be divisible by \(3\).

The sum of the digits in \(98232x4567y\) is:
\[ 9 + 8 + 2 + 3 + 2 + x + 4 + 5 + 6 + 7 + y \]
Substituting \(y = 5\):
\[ 9 + 8 + 2 + 3 + 2 + x + 4 + 5 + 6 + 7 + 5 = 51 + x \]

Now, we need \(51 + x\) to be divisible by \(3\). Since \(51\) is already divisible by \(3\), \(x\) must also be divisible by \(3\).

The possible values for \(x\) that are digits (0 to 9) and divisible by \(3\) are \(0, 3, 6, 9\).

### Step 3: Calculate \(6x - 5y\) for each possible \(x\)
Now we will calculate \(6x - 5y\) for each possible value of \(x\):

1. **If \(x = 0\)**:
\[ 6(0) - 5(5) = 0 - 25 = -25 \]

2. **If \(x = 3\)**:
\[ 6(3) - 5(5) = 18 - 25 = -7 \]

3. **If \(x = 6\)**:
\[ 6(6) - 5(5) = 36 - 25 = 11 \]

4. **If \(x = 9\)**:
\[ 6(9) - 5(5) = 54 - 25 = 29 \]

### Conclusion
The possible values for \(6x - 5y\) are \(-25, -7, 11, 29\). Among these, the only value that matches the options provided in the question is \(29\).

Thus, the final answer is:
\[ \boxed{29} \]