find th sum of digit of least of 7 digits no. that the leaves same remainder 6 when divided by 11,13,17,23.

The smallest number which leaves same remainder 6 when divided by 11,13,17,23 is LCM(11,13,17,23)+6 = 55913+6 = 55919.
All the numbers of the form 55913k+6 leave the same remainder 6 when divided by 11,13,17,23.
We now have to find the smallest 7-digit number of the form 55913k+6.
The smallest such number is when k = 18.
The number is 55913*18+6 = 1006434+6 = 1006440.
Sum of digits of the above number = 1+6+4+4 = 15.