If a 9-digit number 389x6378y is divisible by 72, then the value of 6x+7y is:

For a number to be divisible by 72, it must be simultaneously divisible by both 8 and 9 (since LCM(8,9)=72).
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.
Let's check for values of y∈[0,9]:

8
780+y
​

When we divide 780 by 8, the remainder is 4 (780=97×8+4).
So, 4+y must be divisible by 8. The only single-digit solution is y=4.

Step 2: Divisibility by 9
A number is divisible by 9 if the sum of its digits is a multiple of 9.

Sum of digits=3+8+9+x+6+3+7+8+y
Substitute y=4:

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

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

6x+7y=6(6)+7(4)=36+28=64