the special property of even odd integer and non integer

[Redhorsep]

Hi,

This question pops up when I try to recreate a question on my own based on the official guide problem (this is what MGMAT recommend you to do to stimulate your creativity lol):

OG problem: If m is an integer, is m odd?
1) m/2 is NOT an even integer (turns out this is insufficient)

So I recreated the statement as: m/2 is NOT an odd integer

(I believe this is sufficient, because if that's the case, m/2 could only be even integer or non-integer, but if the result is a non-integer, then m wouldn't be an integer in the first case, thus m/2 must be an even integer and m must be an even integer)

Please help me answer this even/odd integer general rule question, it would be nice if you can show me some examples to demonstrate the logic behind this nature, thanks!

[goalevan]

This would not be sufficient either, since m could still take either an even or odd value.

1) m/2 is NOT an odd integer

m/2 = k, or m = 2k, where k is not an odd integer. It is important to understand that k is not necessarily an integer at all.

Think of a few values of k that are either even integers or non-integers:

Even:
k=2, m=4
k=6, m=12
k=14, m=28

We can generalize that for an even integer k, m will always be a multiple of 2 and thus even because m = 2k. In fact, any integer k would make m even.

But, if we have non-integers:
k=1.5, m=3
k=5.5, m=11
k=21.5, m=43

In this case k is not an odd integer, but m is an odd integer.

[gmatboost]

Great explanation already provided.
I just wanted to point out that the incorrect assumption in the initial reasoning was:

but if the result is a non-integer, then m wouldn't be an integer in the first case,

Be careful not to assume that if n/2 is not an integer, then n is not either. Same for "sqrt(n) is not an integer."