Given:
The standard deviation of the numbers 0, 1, 2, 3, ..., 9 is K.

To find:
The standard deviation of the numbers 10, 11, 12, 13, ..., 19.

Solution:
To find the standard deviation of a set of numbers, we need to follow these steps:

Step 1: Find the mean:
The mean of a set of numbers is found by adding up all the numbers and dividing by the total count.
For the first set of numbers, the mean is (0 + 1 + 2 + 3 + ... + 9) / 10 = 4.5.

Step 2: Find the difference from the mean:
For each number in the set, subtract the mean from it.
For the first set of numbers, the differences are (-4.5, -3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5, 4.5).

Step 3: Square the differences:
Square each of the differences obtained in Step 2.
For the first set of numbers, the squared differences are (20.25, 12.25, 6.25, 2.25, 0.25, 0.25, 2.25, 6.25, 12.25, 20.25).

Step 4: Find the variance:
The variance is found by taking the average of the squared differences.
For the first set of numbers, the variance is (20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25) / 10 = 9.

Step 5: Find the standard deviation:
The standard deviation is the square root of the variance.
For the first set of numbers, the standard deviation is sqrt(9) = 3.

Conclusion:
From the given information, we know that the standard deviation of the first set of numbers is K. Since the second set of numbers is obtained by adding 10 to each number in the first set, the mean of the second set will be the mean of the first set plus 10. Therefore, the mean of the second set is 4.5 + 10 = 14.5.

The differences, squared differences, variance, and standard deviation of the second set will be the same as the first set since only the values have changed, not the pattern. Therefore, the standard deviation of the second set is also K. Hence, the correct answer is option B: K.