The least values of x and y so that 7x342y is divisible y 88 are

Options

A. 4, 4
B. 4, 3
C. 5, 6
D. 6, 7

Answer

To find the least values of \( x \) and \( y \) such that the number \( 7x342y \) is divisible by \( 88 \), we need to ensure that it is divisible by both \( 8 \) and \( 11 \) since \( 88 = 8 \times 11 \).

### Step 1: Check divisibility by 8
A number is divisible by \( 8 \) if the last three digits form a number that is divisible by \( 8 \). The last three digits of our number are \( 42y \).

To find \( y \), we need \( 42y \) to be divisible by \( 8 \).

- Let's check the values of \( y \) from \( 0 \) to \( 9 \):
- For \( y = 0 \): \( 420 \div 8 = 52.5 \) (not divisible)
- For \( y = 1 \): \( 421 \div 8 = 52.625 \) (not divisible)
- For \( y = 2 \): \( 422 \div 8 = 52.75 \) (not divisible)
- For \( y = 3 \): \( 423 \div 8 = 52.875 \) (not divisible)
- For \( y = 4 \): \( 424 \div 8 = 53 \) (divisible)
- For \( y = 5 \): \( 425 \div 8 = 53.125 \) (not divisible)
- For \( y = 6 \): \( 426 \div 8 = 53.25 \) (not divisible)
- For \( y = 7 \): \( 427 \div 8 = 53.375 \) (not divisible)
- For \( y = 8 \): \( 428 \div 8 = 53.5 \) (not divisible)
- For \( y = 9 \): \( 429 \div 8 = 53.625 \) (not divisible)

Thus, the least value of \( y \) that makes \( 42y \) divisible by \( 8 \) is \( y = 4 \).

### Step 2: Check divisibility by 11
Next, we need to check for divisibility by \( 11 \). The rule for divisibility by \( 11 \) states that the difference between the sum of the digits at odd positions and the sum of the digits at even positions must be \( 0 \) or a multiple of \( 11 \).

The digits in \( 7x3424 \) (since we found \( y = 4 \)) are:
- Odd positions: \( 7, 3, 4 \) (1st, 3rd, 5th)
- Even positions: \( x, 2, 4 \) (2nd, 4th, 6th)

Calculating the sums:
- Sum of digits at odd positions: \( 7 + 3 + 4 = 14 \)
- Sum of digits at even positions: \( x + 2 + 4 = x + 6 \)

Now we set up the equation:
\[ 14 - (x + 6) = 14 - x - 6 = 8 - x \]
For \( 7x3424 \) to be divisible by \( 11 \), \( 8 - x \) must be \( 0 \) or a multiple of \( 11 \).

Setting \( 8 - x = 0 \):
\[ x = 8 \]

### Conclusion
Thus, the least values of \( x \) and \( y \) that make \( 7x342y \) divisible by \( 88 \) are:
- \( x = 8 \)
- \( y = 4 \)

### Final Answer
The least values of \( x \) and \( y \) are \( x = 8 \) and \( y = 4 \).