Let x be the least number divisible by 13, such that when x is divided by 4, 5, 6, 7, 8 and 12, the remainder in each case is 2. The sum of the digits of x is:

A number leaving a remainder of 2 when divided by 4, 5, 6, 7, 8, and 12 must be 2 more than the least common multiple of these divisors. The least common multiple is 840, so the number must be a multiple of 13 in the form 840k + 2. Testing small multipliers gives k = 2, which results in x = 1682, and since 1682 = 13 times 129, it is the correct number. The sum of its digits is 1 + 6 + 8 + 2 = 11.

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