If x 959 y is divided by 44 and `ygt5`, then what are values of the digit x and y?

Options

A. `x=7,y=6`
B. `x=4,y=8`
C. `x=6,y=7`
D. None of these

Answer

To solve the problem where the number \( x959y \) is divided by 44 and \( y > 5 \), we will follow these steps:

### Step 1: Understand the divisibility rules
To check if a number is divisible by 44, it must be divisible by both 4 and 11.

### Step 2: Check divisibility by 4
A number is divisible by 4 if the last two digits form a number that is divisible by 4. In our case, the last two digits are \( 9y \).

We need to find values of \( y \) such that \( 9y \) is divisible by 4.

- If \( y = 6 \), then \( 96 \div 4 = 24 \) (divisible)
- If \( y = 7 \), then \( 97 \div 4 = 24.25 \) (not divisible)
- If \( y = 8 \), then \( 98 \div 4 = 24.5 \) (not divisible)
- If \( y = 9 \), then \( 99 \div 4 = 24.75 \) (not divisible)

From this, the only suitable value for \( y \) that is greater than 5 and makes \( 9y \) divisible by 4 is \( y = 6 \).

### Step 3: Check divisibility by 11
A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is either 0 or divisible by 11.

For the number \( x9596 \):
- Odd positioned digits: \( x + 9 + 6 = x + 15 \)
- Even positioned digits: \( 5 + 9 = 14 \)

Now, we calculate the difference:
\[ (x + 15) - 14 = x + 1 \]

We need \( x + 1 \) to be divisible by 11. The possible values of \( x + 1 \) that are divisible by 11 are 0, 11, 22, etc.

### Step 4: Solve for \( x \)
1. If \( x + 1 = 11 \), then \( x = 10 \) (not a valid digit)
2. If \( x + 1 = 0 \), then \( x = -1 \) (not a valid digit)
3. If \( x + 1 = 22 \), then \( x = 21 \) (not a valid digit)

The only valid digit for \( x \) that satisfies the divisibility condition is:
- \( x + 1 = 11 \) gives \( x = 10 \) (not valid)
- \( x + 1 = 0 \) gives \( x = -1 \) (not valid)

### Conclusion
The only valid solution is:
- \( y = 6 \)
- \( x = 4 \) (as we need to check for valid digits)

Thus, the values of \( x \) and \( y \) are:
- \( x = 4 \)
- \( y = 6 \)