q can be 1.
If q =2, number would be of the form 28n + 2 which is a multiple of 2.
Similarly, when q = 4, number would be of the form 28n + 4 which is again a multiple of 2. Any number of the form 28n + an even number will be a multiple of 2.
When q = 7, number would be of the form 28n + 7 which is a multiple of 7.
So, the only remainders possible are remainders that share no factors with 28. Or numbers that are co-prime to 28.
There is a formula for this and a shorter way of finding the number of numbers co-prime to a given natural number. A more detailed discussion on this is provided here and here.
1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25 and 27. q can take 12 different values.
Correct Answer: 12 values