Each statement alone is sufficient. Is C the right answer? I may be missing out on something but here is what i could think of -

If 2n is a factor of x then x is even. Hence k and m both are even.

1. If k > m then the 2-height of k will be greater than the 2-height of m.
(k=6, m=4; 2-height of k=3 and 2-height of m=2). This is true for all values of k and m satisfying the given condition.

2.If m/k is an even integer then
m/k >= 2 => m > k.
2-height of m > 2-height of k

Unfortunately the answer seems different.

Can anyone help us?

Let me know if below analysis looks OK.

The 2-height function is actually applicable to even numbers & provides half value for an even number.
It can be simplified as:

n (of K) = K/2 (if K is a positive even integer)

For odd numbers n is not an integer hence not defined.

Hence
1. if K > M, if either K or M, or both are odd then no answer
2. M/K = even integer: M = (2A)K so n of M is always greater or equal to n of K.

Hence 2 alone can help in deciding; so B is the answer.

Let me know if below analysis looks OK.

The 2-height function is actually applicable to even numbers & provides half value for an even number.
It can be simplified as:

n (of K) = K/2 (if K is a positive even integer)

For odd numbers n is not an integer hence not defined.

Hence
1. if K > M, if either K or M, or both are odd then no answer
2. M/K = even integer: M = (2A)K so n of M is always greater or equal to n of K.

Hence 2 alone can help in deciding; so B is the answer.

Thanks - yes answer is B.