The greatest number which will divides: 4003, 4126 and 4249, leaving same remainder in each case:

Rule- Greatest number with which if we divide P, Q, R and it leaves same remainder in each case. Number is of form = HCF of (P - Q), (P - R)
Therefore, HCF of (4126 - 4003), (4249 - 4003) = HCF of 123, 246 = 41.[Taken for Positive result].

Detailed Explanation:
The numbers can be written as,
4003 = AX + P where P = Remainder
4126 = BX + P
4249 = CX + P
(B - A) × X = 123
(C - B) × X = 246
Thus the X is factor of 123 and 246