### Instruction: 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?### Response: I need to find the sum of the first ten terms of any arithmetic sequence whose terms are positive integers.
I recall that the formula for the sum of an arithmetic series is S_n = n/2 (a_1 + a_n), where n is the number of terms, a_1 is the first term, and a_n is the last term.
In this case, n = 10, so S_10 = 10/2 (a_1 + a_10) = 5 (a_1 + a_10).
I notice that this sum is always divisible by 5, regardless of the values of a_1 and a_10.
I wonder if there is any larger number that must also divide this sum for any positive integer values of a_1 and a_10.
I realize that the answer depends on the common difference of the arithmetic sequence, which I will call d.
If d is even, then a_1 and a_10 have the same parity (they are both even or both odd), and therefore their sum is even.
In that case, S_10 is also even, and hence divisible by 10.
If d is odd, then a_1 and a_10 have opposite parity (one is even and the other is odd), and therefore their sum is odd.
In that case, S_10 is not even, and hence not divisible by 10.
Therefore, the largest number that must divide S_10 for any positive integer values of a_1 and a_10 is 5.