If an 11-digit number 765y88436x6 is divisible by 72, and x assumes the largest value, then what is the value of (x - y)?

Options

A. 4
B. 5
C. 8
D. 6

Answer

To determine the values of \( x \) and \( y \) in the number \( 765y88436x6 \) such that it is divisible by 72, we need to check two conditions: divisibility by 8 and divisibility by 9.

### 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. In our case, the last three digits are \( 36x6 \).

To find the values of \( x \) that make \( 36x6 \) divisible by 8, we can check each digit from 0 to 9 for \( x \):

- \( x = 0 \): \( 360 \div 8 = 45 \) (divisible)
- \( x = 1 \): \( 361 \div 8 = 45.125 \) (not divisible)
- \( x = 2 \): \( 362 \div 8 = 45.25 \) (not divisible)
- \( x = 3 \): \( 363 \div 8 = 45.375 \) (not divisible)
- \( x = 4 \): \( 364 \div 8 = 45.5 \) (not divisible)
- \( x = 5 \): \( 365 \div 8 = 45.625 \) (not divisible)
- \( x = 6 \): \( 366 \div 8 = 45.75 \) (not divisible)
- \( x = 7 \): \( 367 \div 8 = 45.875 \) (not divisible)
- \( x = 8 \): \( 368 \div 8 = 46 \) (divisible)
- \( x = 9 \): \( 369 \div 8 = 46.125 \) (not divisible)

Thus, the possible values for \( x \) that make \( 36x6 \) divisible by 8 are \( 0 \) and \( 8 \).

### Step 2: Check divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits of \( 765y88436x6 \) is:

\[ 7 + 6 + 5 + y + 8 + 8 + 4 + 3 + 6 + x + 6 = 43 + y + x \]

We need to check the sums for \( x = 0 \) and \( x = 8 \):

1. **If \( x = 0 \)**:
\[ 43 + y + 0 = 43 + y \]
We want \( 43 + y \equiv 0 \mod 9 \). The remainder of 43 when divided by 9 is 7, so we need:
\[ 7 + y \equiv 0 \mod 9 \implies y \equiv 2 \mod 9 \]
The possible values for \( y \) are \( 2 \) or \( 11 \) (but \( y \) must be a single digit, so \( y = 2 \)).

2. **If \( x = 8 \)**:
\[ 43 + y + 8 = 51 + y \]
We want \( 51 + y \equiv 0 \mod 9 \). The remainder of 51 when divided by 9 is 6, so we need:
\[ 6 + y \equiv 0 \mod 9 \implies y \equiv 3 \mod 9 \]
The possible values for \( y \) are \( 3 \) or \( 12 \) (but again, \( y \) must be a single digit, so \( y = 3 \)).

### Step 3: Determine the largest value of \( x \)
From our calculations, we have:
- If \( x = 0 \), then \( y = 2 \).
- If \( x = 8 \), then \( y = 3 \).

Since we want the largest value of \( x \), we take \( x = 8 \) and \( y = 3 \).

### Step 4: Calculate \( x - y \)
Now, we can find \( x - y \):
\[ x - y = 8 - 3 = 5 \]

### Final Answer
The value of \( x - y \) is \( 5 \).

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