### Instruction: How many positive integers less than 1000 are congruent to 6 (mod 11)?### Response: To be congruent to 6 (mod 11), a number must have a remainder of 6 when divided by 11.
I can write this as n = 11k + 6, where k is a non-negative integer.
To find the possible values of k, I need to make sure that n is less than 1000.
This means that 11k + 6 < 1000, or 11k < 994.
Dividing both sides by 11, I get k < 90.4.
Since k has to be an integer, the largest possible value of k is 90.
The smallest possible value of k is 0, since k has to be non-negative.
So there are 91 possible values of k, from 0 to 90.
Each value of k corresponds to a unique value of n that is congruent to 6 (mod 11) and less than 1000.
Therefore, there are 91 positive integers less than 1000 that are congruent to 6 (mod 11).