Explanation

Remainders 5, 7, 11 are each one less than the divisors 6, 8, 12, so N + 1 must be a common multiple of 6, 8, 12: N + 1 = 24k, i.e. N = 24k - 1. For N to be a multiple of 13, 24k - 1 = 0 mod 13 gives 11k = 1 mod 13, whose least solution is k = 6. Then N = 144 - 1 = 143 = 11 x 13, and the sum of its digits is 1 + 4 + 3 = 8. Hence option (d).