If two numbers 121 and 199 are divided by two digits number then leaves the same remainder in each case.  How many two digits numbers of such type are possible?

4

Required number = HCF and factors of HCF of (121 - x), (199 - x)

                             121           199

                               ↓               ↓

                         Diff. = 78 (HCF)

Required numbers = 78 and two digit factors of 78

                           = 78, 39, 26, 13

                   ∴ 4 numbers are possible