A boy divided the numbers 7654, 8506 and 9997 by a certain largest number and he gets the same remainder in each case. What is the common remainder?

If the remainder is same in each case and remainder is not given,
HCF of the differences of the numbers is the required largest number.

9997 - 7654 = 2343
9997 - 8506 = 1491
8506 - 7654 = 852
Hence, the greatest number which divides 7654, 8506 and 9997 and leaves same remainder
= HCF of 2343, 1491, 852
= 213
Now we need to find out the common remainder.
Take any of the given numbers from 7654, 8506 and 9997, say 7654
7654/213 = 35,
remainder = 199.