If a nine-digit number 785x3678y is divisible by 72, then the value of (7x−5y) is:

For a number to be divisible by 72, it must be divisible by both 8 and 9 since gcd(8,9)=1.
Step 1: 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 78y by 8:

78y=780+y=8×97+(4+y)
For (4+y) to be divisible by 8 where 0≤y≤9, we must have:

y=4
Step 2: Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

Sum of digits=7+8+5+x+3+6+7+8+y
Substitute y=4:

Sum=7+8+5+x+3+6+7+8+4=48+x
For (48+x) to be divisible by 9, the nearest multiple of 9 greater than 48 is 54.

48+x=54⟹x=6
Step 3: Calculate the value of (7x−5y)

7x−5y=7(6)−5(4)=42−20=22