What is the largest number that, when dividing 697, 909, and 1227, leaves the same remainder in each case?

Explanation

To find the greatest number that yields the same remainder when dividing 697, 909, and 1227, we calculate the differences between the numbers: 909 - 697 = 212, 1227 - 909 = 318, and 1227 - 697 = 530. The largest number dividing all these differences exactly is the greatest common divisor (GCD) of 212, 318, and 530, which is 106.