Find the greatest number that divides 103, 199 and 343, leaving the SAME remainder in each case.

When the common remainder is not stated, work with the differences: 199 - 103 = 96, 343 - 199 = 144 and 343 - 103 = 240. HCF(96, 144, 240) = $2^4\times$ 3 = 48, and indeed each of 103, 199 and 343 leaves remainder 7 on division by 48. Answering 96 uses only one difference (and 96 does not divide 144); 24 and 12 divide all the differences but are not the greatest.