Question 1210235
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Let S be a set of distinct integers. What is the smallest number of elements that S must contain, 
to ensure that S has a nonempty subset, where the sum of the elements in the subset is divisible by 2?
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I will treat the problem differently.


First,  the problem says  " Let  S  be a set of distinct integers."
So,  I will assume that the number of elements in  S  is at least  2  (two),  in order for the term 
"distinct integers"  would make its natural sense.


Second,  I will assume that when the problem says  " the sum of the elements in the subset ",
it means that at least two elements of the subset are involved/included into the sum;  
otherwise it is like &nbsp;" dance tango alone ".  &nbsp;<---> &nbsp;Alhough it is possible, &nbsp;but it is unnatural.



Then the answer to the problem's question is 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;"the smallest number of elements that &nbsp;S &nbsp;must contain is &nbsp;3 &nbsp;(three)".



<pre>
Indeed, if the set S contains three or more distinct integers, then inevitably 

    EITHER there is a pair of two distinct even integers in S, giving the even sum,

    OR     there is a pair of two distinct odd integers in S, giving the even sum.


So, any set S containing three or more distinct integers, satisfies the condition.


On the contrary, the set of two distinct integers may have one even number and one odd number;
then the sum of these two integer numbers is an odd integer.
So, such a set S of two integers of different parity fails the condition.


Thus, if to treat the problem this way, then the answer is 

        "the smallest number of elements that S must contain is 3 (three)".
</pre>

Solved.