{S(D)} denotes a strictly ascending arithmetic sequence whose first term is 1 and the common difference is D, where D is a positive integer.

For how many values of D does {S(D)} contain the term 2017?

For both 1 and 2017 to occur in an integer valued arithmetic sequence, the common difference D must be a factor of the difference 2017-1=2016. 2016 = 2^5*3^2*7^1, which means that 2016 has (5+1)*(2+1)*(1+1) = 36 factors. That is the answer: 36 values of D (the factors of 2016) exist such that {S(D)} contains 2017.