Understanding Divisibility Rules

To solve this problem, we need to understand the divisibility rules for 88. A number is divisible by 88 if and only if it is divisible by both 8 and 11, because 8 and 11 are coprime (their greatest common divisor is 1) and their product is 88.

Applying Divisibility by 8 Rule

Rule for 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

In the given 10-digit number, 643x1145y2, the last three digits form the number 5y2.

We need to find the possible values for the digit 'y' (where y can be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) such that 5y2 is divisible by 8.

Let's check the values:

• 502 ÷ 8 = 62 remainder 6
• 512 ÷ 8 = 64 (So, y=1 is possible)
• 522 ÷ 8 = 65 remainder 2
• 532 ÷ 8 = 66 remainder 4
• 542 ÷ 8 = 67 remainder 6
• 552 ÷ 8 = 69 (So, y=5 is possible)
• 562 ÷ 8 = 70 remainder 2
• 572 ÷ 8 = 71 remainder 4
• 582 ÷ 8 = 72 remainder 6
• 592 ÷ 8 = 74 (So, y=9 is possible)

The possible values for 'y' are {1, 5, 9}.

Applying Divisibility by 11 Rule

Rule for 11: A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or a multiple of 11.

The given number is 643x1145y2.

Let's find the sums:

Sum of digits at odd places (1st, 3rd, 5th, 7th, 9th):

$$ S_{odd} = 6 + 3 + 1 + 4 + y = 14 + y $$

Sum of digits at even places (2nd, 4th, 6th, 8th, 10th):

$$ S_{even} = 4 + x + 1 + 5 + 2 = 12 + x $$

The difference is $ S_{odd} - S_{even} = (14 + y) - (12 + x) = 2 + y - x $.

This difference must be a multiple of 11. Since x and y are single digits (0-9), the value of $2 + y - x$ can range from $2+0-9 = -7$ to $2+9-0 = 11$. The only multiples of 11 in this range are 0 and 11.

So, we have two possibilities:

1. $2 + y - x = 0 \implies x = y + 2$
2. $2 + y - x = 11 \implies x = y - 9$

Finding x for the Largest Value of y

The question requires us to find the value of the expression $(2x - 3y)$ specifically for the largest value of y.

From our analysis of the divisibility by 8 rule, the possible values for y are {1, 5, 9}.

The largest value among these is $y=9$.

Now, we use the divisibility by 11 conditions to find the corresponding value of 'x' when $y=9$.

• Check condition 1: $x = y + 2$. If $y=9$, then $x = 9 + 2 = 11$. This is not possible because 'x' must be a single digit (0-9).
• Check condition 2: $x = y - 9$. If $y=9$, then $x = 9 - 9 = 0$. This is a valid single digit.

Therefore, for the largest possible value of y, which is 9, the value of x must be 0.

Calculating the Expression (2x - 3y)

We need to calculate the value of $(2x - 3y)$ using the values we found: $x=0$ and $y=9$.

Substitute these values into the expression:

$$ (2x - 3y) = (2 \times 0 - 3 \times 9) $$

First, perform the multiplications:

$$ (2 \times 0) = 0 $$

$$ (3 \times 9) = 27 $$

Now, perform the subtraction:

$$ (2x - 3y) = 0 - 27 $$

$$ (2x - 3y) = -27 $$