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

For a number to be divisible by 72, it must be divisible by both 8 and 9.
Step 1: Divisibility by 8
The last three digits of the number must be divisible by 8. The last three digits are 6x6.
Testing values for x from 9 downwards (since we want the largest value of x):

If x=9, 696÷8=87 (Divisible)

If x=5, 656÷8=82 (Divisible)

If x=1, 616÷8=77 (Divisible)

Since x assumes the largest value, x=9.

Step 2: Divisibility by 9
The sum of the digits must be divisible by 9.
Number = 765y8843696 (substituting x=9)
Sum of digits = 7+6+5+y+8+8+4+3+6+9+6=62+y

For (62+y) to be divisible by 9, the nearest multiple of 9 greater than or equal to 62 is 63.

62+y=63⟹y=1
Step 3: Find the value of (x−y)

x−y=9−1=8