Is x odd? (1) x/2 is not an integer (2) 2x + 3 is odd.

Consider (1):

For all even integers, x/2 is an integer. if x/2 is not an integer, either x is not an integer, or x is an odd integer. Since odd is defined only for integers, x may or may not be odd.

Consider (2):
2x + 3 is odd

For 2x+3 to be odd, 2x must be even and hence x must be an integer. (eg: 1.5*2 = 3 (ODD)) However, x can be both even / odd since 2x is even for any integer.

If we combine this inference with (1), we know that x is an integer and x is odd. Hence C.

option B -- both yes/no -- 2x+3=odd where x can odd or even .
option A --x/2 = not integer
if x is even then x/2 integer
if x is odd then x/2 not an integer
Hence A .
Ans

is x odd?

Statement 1

x may not be an integer

3/2 is not an integer (x=3)
.5/2 is also not an integer (x=.5)

Not Sufficient

Statement 2

2x+3 is odd
2(2) + 3 is also odd
2*(3) +3 is also odd

Not Sufficient as in both cases its even + odd=odd

Both together will be sufficient

This is because 2x has to be an integer and 2x can only be even to make the sum 2x+3 odd
Therefore x can only be an integer ( it cant be values such as 2(.5)=1 or 2(3.5)=7 which will make sum 2x+3 even)
Since x/2 is not an integer, x can only be an odd integer

Hence IMO C

KAPLAN OFFICIAL SOLUTION:

When I elicit permissible numbers for Statement (1), students often give me numbers like 3, 5, and 7. Sure enough, all of these produce non-integers when divided by 2, and all are odd, so students lean towards sufficient. But wait! Who said x had to be an integer? Certainly a number like 6.4289 would yield a non-integer when divided by 2, so it’s permissible, and it’s not odd. Sometimes Yes, Sometimes No: Insufficient!

Similarly, we have to recognize that odds and evens are both permissible for Statement (2), and therefore Statement (2) is insufficient. [By the way, put the statements together and you’ll find that (C) is correct. See why?] So we have to consider odds vs. evens and integers vs. non-integers in such number properties questions.

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