Let x be the smallest number greater than 600 which gives the remainders 2, 3 and 4, when divided by 5, 6 and 7, respectively. The sum of digits of x is:

Here we need to take the LCM of 5, 6 and 7. Because the required number is divided by these and leaves some remainders.

5 = $$5\times1$$

6 = $$3\times2\times1$$

7 = $$7\times1$$

LCM of 5, 6 and 7 = $$7\times5\times3\times2\times1$$

= 210

From the given information, we know that x is the smallest number greater than 600. So 210 is multiplied by 3 to obtain a number that is greater than 600.

Here the difference between remainders and their respective number from which these are divided is the same.

5-2 = 3 

6-3 = 3

7-4 = 3

hence the required number = x = $$210\times3-3$$

= 630-3

= 627

the sum of digits of x = 6+2+7

= 15