The longest streak of consecutive numbers, NONE of whose digit sums is a multiple of 7, is 12 in a row. For instance:

994, digit sum = 9+9+4 = 22, remainder after division by 7 = 1

995, digit sum = 9+9+5 = 23, remainder after division by 7 = 2

996, digit sum = 24, remainder after division by 7 = 3

997, digit sum = 25, remainder after division by 7 = 4

998, digit sum = 26, remainder after division by 7 = 5

999, digit sum = 27, remainder after division by 7 = 6

1000, digit sum = 1, remainder after division by 7 = 1

1001, digit sum = 2, remainder after division by 7 = 2

1002, digit sum = 3, remainder after division by 7 = 3

1003, digit sum = 4, remainder after division by 7 = 4

1004, digit sum = 5, remainder after division by 7 = 5

1005, digit sum = 6, remainder after division by 7 = 6

By the same notion, the length of the longest streak of consecutive numbers, NONE of whose digital sums is a multiple of 13, happens to be an exact multiple of 13 itself.

How long is the streak, and can you find an example?