If the 9-digit number 83P93678Q is divided by 72, then what is the value of $\sqrt{P^2 +Q^2 +12}$ ?

8

If the 9-digit number 83P93678Q is divided by 72

Then it is also divisible by 8 and 9.

According to the divisibility of 8 the last digit of the number is divisible by 8 so, 78Q is divisible by 8 for the value of Q = 4

And according to the divisibility of 9 the sum of the digits of the number is divisible by 9.

83P936784 = 8 + 3 + P + 9 + 3 + 6 + 7 + 8 + 4 = 48 + P 

48 + P = 54 ( 54 is divisible by 9)

P = 6

$\sqrt{P^2 +Q^2 +12}$ = $\sqrt{6^2 +4^2 +12}$ = 8