take few odd numbers as -3,-1, 1,3 and work out what happens to m/2 it gives answer which says that m/2 is not an even integer and m can be -3 , -1,1 ,3
again take even number -2,0,2 and work out m/2 which gives an answer that m/2 is not an even integer. This means that m can be even interger and at the same time odd integer hence statement (1) is not sufficient.

statement (2) is sufficient

hence the answer is B

what you're missing is the interpretation of the statement "m/2 is not an even integer". It means just that: m/2 can be anything, as long as it's not an even integer. It's not the same as "m/2 is odd": if m/2 is a fraction such as 3/2, then m/2 is not an even integer - it's not an integer at all, but that's fine, according to statement (1).

It's these little end cases and differences in interpretation that makes DS questions so tricky. Your job here is almost always to think of the opposite scenario, and how the statements allow this opposite scenario to happen: "m/2 is not an even integer" is not the same as "m/2 is odd", and there's a reason why they used one phrasing and not the other. Figure out why!