What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?
I know that the sum of the first n terms of an arithmetic sequence is given by the formula S_n = n(a_1 + a_n) / 2, where a_1 is the first term and a_n is the nth term. So, to find the sum of the first ten terms, I can plug in n = 10 and get S_10 = 10(a_1 + a_10) / 2. I can simplify this to S_10 = 5(a_1 + a_10).
[*. So, the sum of the first ten terms is always a multiple of 5, regardless of the values of a_1 and a_10. Therefore, the greatest positive integer that must divide S_10 is 5. 