SOLUTION: Prove that given any set of $17$ integers, there exist nine of them whose sum is divisible by $2.$

Question 1208363: Prove that given any set of $17$ integers, there exist nine of them whose sum is divisible by $2.$
Found 3 solutions by mccravyedwin, Edwin McCravy, ikleyn:
Answer by mccravyedwin(410)   (Show Source): You can put this solution on YOUR website!

This is false because If we have 17 odd positive integers, no 9 of them will have an even sum. Edwin

Answer by Edwin McCravy(20067)   (Show Source): You can put this solution on YOUR website!

The sum of an odd number of odd numbers is always odd, never even 9 is an odd number, so if all 17 were odd, the statement would be false. However, a question you might ask is whether a set of 17 odd positive integers is the ONLY counterexample, or are there any others? Certainly 16 odds and 1 even is not a counterexample because 8 odds and 1 even would be 9 with an even sum. Edwin

Answer by ikleyn(52943)   (Show Source): You can put this solution on YOUR website!
.
Prove that given any set of $17$ integers, there exist nine of them whose sum is divisible by $2.$
~~~~~~~~~~~~~~~~~~~~~~~~~~~

As it is worded, printed, posted and presented, it is TOTALLY and FATALLY wrong statement.

Take, for example, 17 integers, all equal to 1, and you will NEVER get 9 addends,
giving an even sum.   Their sum always will be 9.

Or, if this example does not convince you, take first 17 odd integer numbers

1, 3, 5, 7, . . . 31, 33.

Using these numbers, you will never get 9 of them with an even sum.

        Where did you get this gibberish ?    What is the source ?

                Just a shamefully low level of Math composing.

////////////////////////////////////

I placed this formulation in my Google browser and requested for search.

It generates first output created by Artificial Intelligence (AI).

It becomes making some reasoning, but from its output,
it is clear that this AI incorrectly understands the problem.

The link is

https://www.google.com/search?q=Prove+that+given+any+set+of+%2417%24+integers%2C+there+exist+nine+of+them+whose+sum+is+divisible+by+%242.%24&rlz=1C1CHBF_enUS1071US1071&oq=Prove+that+given+any+set+of+%2417%24+integers%2C+there+exist+nine+of+them+whose+sum+is+divisible+by+%242.%24&gs_lcrp=EgZjaHJvbWUyBggAEEUYOdIBCTE4MTFqMGoxNagCCLACAQ&sourceid=chrome&ie=UTF-8

On the way, this AI generates a blatant nonsense.

At the very end, it confesses that

Generative AI is experimental. Answers might be inaccurate or misleading. Double-check for accuracy.

It is precisely the case when this AI works INCORRECTLY.

Still is not trained to solve such problems.

Had it worked in a right way, it should/(it must) recognize that the problem is posed INCORRECTLY.

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