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:

From the given 4,5,6,7,8,12 

then $$4 = 2 \times 2 $$ , 5 = 5 , 6 = $$2 \times 3 $$, 7 = 7, 8 = $$ 2\times 2 \times 2  $$ , 12 = $$ 2 \times 2 \times 3 $$ 

then LCM = $$ 2 \times 2 \times 2 \times 5 \times 3 \times 7 $$

$$\Rightarrow 840 $$ 

so  the number = 840 k + 2 (according to question)

which divisible by 13 

put k = 1 number =842 (not divisible by 13) 

k=2 number = 1682 (not divisible )

k= 3 number = 2522 (which is divisible)

so $$ x = 2522 $$

sum of digits = 2+5+2+2 = 11 Ans