Largest x that divides at least one sum of a pair of N numbers?

For x = 2 we would require at least 3 natural numbers to be guaranteed that there

is at least one pair whose sum is divisible by 2. For x > 2 then all the N elements

of the set could be 1 (mod x), which would make any sum 2 (mod x) and hence

not divisible by x. So for N >= 3 the answer would be x = 2, and for N = 2 the

answer would be just 1, since the two numbers could be 0 (mod 2) and 1 (mod 2).

(The N = 1 case wouldn't make much sense, as there are no pairs of numbers

to add. :) )

Edit: Oh darn. This is more complicated than I originally thought. If all the

N elements of the set were congruent to 1 (mod x) for some x, say 3, then

the sum of any two would not be divisible by 3, but this might guarantee

that there is some larger x that would divide one of the sums. I need to

give this some more thought...... My gut is telling me x = N - 1, so I'll see

if I can prove that to be correct.

Edit #2: For N = 2 the above analysis stands.

For N = 3, we can have two numbers congruent to 1 (mod 3) and one congruent

to 0 (mod 3) and thus have no sums be divisible by 3. But when we look at the

numbers modulo 2 then we would have to have at least two of the numbers

either 1 (mod 2) or 0 (mod 2), giving us at least one sum that is divisible by 2.

For N = 4 we can have 3 numbers congruent to 1 (mod 4) and one congruent

to 0 (mod 4) and thus have no sums divisible by 4. But when we look at the

4 numbers modulo 3 then we have the following cases:

(i) if one number is 0 (mod 3) then none of the others can be, in which case

we have 3 numbers that are either 1 or 2 (mod 3). If one of these is 1 (mod 3)

then none of the others can be 2 (mod 3), which would leave us with 3 numbers

that are congruent to 1 (mod 3). So in this case we can avoid any sum being

divisible by 3, but we have opened ourselves up to the necessity that some

sum must be divisible by 4, (or perhaps a higher value).

Maybe my original answer wasn't so far off after all. I'll have to sleep on it.

Edit #3: It looks like I was starting the 'over-think' the question in my edits.

My original answer should suffice, but I'll leave my edits intact for your

consideration and amusement. :)