Let x be the least number which when divided by 8, 9, 12, 14 and 36 leaves a remainder of 4 in each case, but x is divisible by 11.  The sum of the digits of x is?

4

LCM of 8, 9, 12, 14 and 36 = 504

(x) required number = 504K + 4 [which is exactly divisible by 11 for certain value of K]

when K = 2, 

Number = 504 × 2 + 4

          x = 1008 + 4 = 1012  which is exactly divisible by 11

Sum of digits of x = 1 + 1 + 2 = 4