document.write( "Question 1210235: 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? \n" ); document.write( "
Algebra.Com's Answer #851718 by ikleyn(52781)\"\" \"About 
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\n" ); document.write( "Let S be a set of distinct integers. What is the smallest number of elements that S must contain,
\n" ); document.write( "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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\n" ); document.write( "\n" ); document.write( "I will treat the problem differently.\r
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\n" ); document.write( "\n" ); document.write( "First,  the problem says  \" Let  S  be a set of distinct integers.\"
\n" ); document.write( "So,  I will assume that the number of elements in  S  is at least  2  (two),  in order for the term
\n" ); document.write( "\"distinct integers\"  would make its natural sense.\r
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\n" ); document.write( "\n" ); document.write( "Second,  I will assume that when the problem says  \" the sum of the elements in the subset \",
\n" ); document.write( "it means that at least two elements of the subset are involved/included into the sum;
\n" ); document.write( "otherwise it is like  \" dance tango alone \".  <--->  Alhough it is possible,  but it is unnatural.\r
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\n" ); document.write( "\n" ); document.write( "Then the answer to the problem's question is \r
\n" ); document.write( "\n" ); document.write( "        \"the smallest number of elements that  S  must contain is  3  (three)\".\r
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document.write( "Indeed, if the set S contains three or more distinct integers, then inevitably \r\n" );
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document.write( "    EITHER there is a pair of two distinct even integers in S, giving the even sum,\r\n" );
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document.write( "    OR     there is a pair of two distinct odd integers in S, giving the even sum.\r\n" );
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document.write( "So, any set S containing three or more distinct integers, satisfies the condition.\r\n" );
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document.write( "On the contrary, the set of two distinct integers may have one even number and one odd number;\r\n" );
document.write( "then the sum of these two integer numbers is an odd integer.\r\n" );
document.write( "So, such a set S of two integers of different parity fails the condition.\r\n" );
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document.write( "Thus, if to treat the problem this way, then the answer is \r\n" );
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document.write( "        \"the smallest number of elements that S must contain is 3 (three)\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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