Problem Analysis:
We are given that a number, when divided by 21, 28, 30, and 35, leaves a remainder of 10 in each case. We need to find the sum of the digits of the smallest such number that is also divisible by 17.

We can solve this problem by finding the LCM (Least Common Multiple) of the given numbers and then finding the smallest multiple of 17 in that range.

Solution:
Let's approach the problem step by step.

Finding the LCM:
To find the LCM of 21, 28, 30, and 35, we can prime factorize each number and then take the highest power of each prime factor.

21 = 3 × 7
28 = 2 × 2 × 7
30 = 2 × 3 × 5
35 = 5 × 7

Taking the highest power of each prime factor:
21 = 3 × 7
28 = 2 × 2 × 7
30 = 2 × 3 × 5
35 = 5 × 7

Since all the prime factors are covered, we can find the LCM by multiplying these highest powers:
LCM = 2 × 2 × 3 × 5 × 7 = 420

Finding the smallest multiple of 17:
To find the smallest multiple of 17 in the range of 420, we can start from 17 and keep adding 420 until we find a multiple that is divisible by 17.

17, 437, 857, 1277, 1597, 2017, 2437, 2857, 3277, 3697, 4117, 4537, 4957, 5377, 5797, 6217, 6637, 7057, 7477, 7897, 8317, 8737, 9157, 9577, 9997, 10417, 10837, 11257, 11677, 12097, 12517, 12937, 13357, 13777, 14197, 14617, 15037, 15457, 15877, 16297, 16717, 17137, 17557, 17977, 18397, 18817, 19237, 19657, 20077, 20597, 21017, 21437, 21857, 22277, 22697, 23117, 23537, 23957, 24377, 24797, 25217, 25637, 26057, 26477, 26897, 27317, 27737, 28157, 28577, 28997, 29417, 29837, 30257, 30677, 31097, 31517, 31937, 32357, 32777, 33197, 33617, 34037, 34457, 34877, 35297, 35717, 36137, 36557, 36977, 37397, 37817, 38237, 38657,