How can an odd set of integers include even numbers as well?

Question

Now, when I see the explanation, it says:

(-11) + (-9) + (-7) + .... + 0 + .... + 7 + 9 + 11 = 0, so the first 23 numbers cancel each other and the sum is 0. Then 13 + 15 = 28. Therefore there are a total of 25 integers.

However, in the question, it is given that it is a set of odd integers. How is it the case that the answer includes the number of even numbers as well?

What is your question? Are you wondering why the designers of this multiple choice test decided to include $24$ among the wrong suggestions? — Hagen von Eitzen, Oct 26 '14 at 11:30
It doesn't. They probably wrote the +0 in the middle instead of (-1+1) and (-3+3) etc, choosing to point out that they total zero (as opposed to zero being present as one of the numbers) — JMoravitz, Oct 26 '14 at 11:32
@JMoravitz Even if you add up all the positive and negative odd numbers till 11, even then you get only 13 numbers and plus two = 15. — Always Learning Forever, Oct 26 '14 at 11:33
Ah, now I get it. The cardinality of ${-11,-9,-7,\ldots, 7,9,11,13,15}$ is in fact much less than $25$ (oh, counting ...) — Hagen von Eitzen, Oct 26 '14 at 11:34

Hagen von Eitzen · Accepted Answer · 2014-10-26T15:41:01.780

5

Without the suggested answers, the question does not have a unique answer. For example, $\{-11, 39\}$ is a set of odd integers that has $-11$ as least member and the sum is $28$ and so is $\{-11,-5,7,37\}$.

In general, if the set had an odd number of elements, the sum would be odd because all summands are odd. That immediately rules out the answer options $11,23,25$.

The minimal sum that $22$ distinct (because the problem statement talks about a set!) odd integers $\ge -11$ could produce is $$-11-9-7-5-3-1+1+3+5+7+9+11+\\+13+15+17+19+21+23+25+27+29+31 \gg28 $$ and with $24$ summands it gets even worse.

Conclusion: Dump that test.

edited Oct 26 '14 at 15:41

answered Oct 26 '14 at 11:39

Hagen von Eitzen

374,180

I would give $-11-9-7-\cdots-1+1+\cdots+29+31$ for the conclusion, if that option was available. – Marc van Leeuwen Oct 26 '14 at 11:48
However you meant "The minimal sum that a set of $22$ odd integers$\geq11$ could produce"; normally a sum of odd integers could contain the same term more than once. But the test indeed says "a set". – Marc van Leeuwen Oct 26 '14 at 11:53
@MarcvanLeeuwen Duely noted – Hagen von Eitzen Oct 26 '14 at 15:41

How can an odd set of integers include even numbers as well?

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