Question 253793: a number is called " even - odd " if it is halfway between an even integer and an odd integer. if x is an even-odd number , which of the following must be true ?
I. 2x is an integer
II 2x is even-odd
III x is halfway between two even integers
A) I only (B) II only (C) I and II only
(D) II and III only (E) I, II, and III
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! pick any odd number.
start with 3.
take a number greater than it that is even.
take a number less than it that is also even.
take 18 and -18
these form the odd-even pairs we will work with.
number halfway between 3 and 18 is 15/2 = 7.5
number halfway between 3 and -18 is 21/2 = 10.5
neither of these number is an integer.
an even-odd number is therefore not an integer.
2 times the number, however, will be an integer.
2 * 7.5 = 15
2 * 10.5 = 21
statement 1 is true.
2 times the number is not even-odd.
can't be, because 2 times the number is an integer and even-odd number is not an integer.
statement 2 is false.
statement 3 says even-odd integer is halfway between 2 even integers.
can't be because an odd integer is halfway between 2 even integers and the even-odd integers is halfway between one of the even integers and the odd integers so it can't be halfway between the 2 even integers.
take the case of 3 and 18
even-odd integer is halfway between them and is equal to 15/2 = 7.5
closest even integer higher is 8.
closest even integer lower is 6
7 is halfway between 6 and 8
7.5 is therefore not.
a number between 2 even integers will always be an integer which is not an odd-even integer.
statement 3 is false.
your answer should be selection A which is statement 1 is true only.
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