If a nine-digit number 385x3678y is divisible by 72, then the value of (y−x) is:

For a number to be divisible by 72, it must be simultaneously divisible by both 8 and 9 (since gcd(8,9)=1).
Step 1: Test for divisibility by 8
A number is divisible by 8 if its last three digits form a number divisible by 8. The last three digits are 78y.
Dividing 780 by 8:

780=8×97+4
To make 78y perfectly divisible by 8, the remaining part must complete a multiple of 8. Thus, 40+y must be divisible by 8, which gives y=4 (since 784÷8=98).

So, y=4.

Step 2: Test for divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
The number is now 385x36784. Let's compute the sum of the digits:

Sum=3+8+5+x+3+6+7+8+4=44+x
For (44+x) to be a multiple of 9, the nearest multiple greater than or equal to 44 is 45.

44+x=45⟹x=1
Step 3: Calculate the value of (y−x)

y−x=4−1=3