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Please upload your work for #1 through #9 as a single PDF file.  To retrieve your work via Gmail with Honor Lock enabled, use the link below. Gmail Inbox If you are using Airdrop, you can ignore this.  You are not able drag and drop the file to Canvas. Click choose a file and locate the file in the finder (mac) or file explorer (windows) window that pops up to locate the file.  

11) Theorem: A group of  kids have a total of  chocolate bars. Then at least one of the kids has fewer than three chocolate bars.A proof by contradiction of the theorem starts by assuming which fact?

1) One way to show that the number  is rational is to show that  , where  and  are integers and  is non-zero. Select a pair of values for  and  to show that  is rational.

5) Select the mistake that is made in the proof given below.Theorem. The product of an even integer and any other integer is even.Proof.Let  be an even integer and  be an arbitrary integer. Since  is even,  for some integer  . Therefore,  for some integer  , which means that  is even.

2) Select thetruestatement.

13) Theorem: For any two real numbers,  and  , such that  ,  .One of the cases in the proof of the theorem says: Since  and  ,  .Select the case that corresponds to this argument.

4) What must be proven to establish that the following statement isfalse:There are two consecutive positive integers that are both odd.

“The Way of the Elders.” A school of Buddhism that emphasizes the monastic ideal.

10) Theorem: The average of any two positive real numbers is also positive.Which facts are assumed and which facts are proven in a proof by contrapositive of the theorem?

Hesiod believed that farmers possessed all of the following virtues except