how can I show this :T contains two numbers such that one divided the other?

Question

Let T ⊂ {1,2,…,2n} where T has n +1 elements. Then T contains two numbers such that one divided the other. [any number could be written uniquely as an odd number times a power of 2, m = (2k – 1)2^j. Consider the function f: T -> {1,2,…n} defined by f(m) = k]

Answer 1

As the hint says, every number is some power of two multiplied by an odd number, or 2^j * s, where s is odd.
Now, lets say you have written every number in the set T like that. What possible values for s have you come up with? They all have to be <= 2n-1, since the whole number is <= 2n. That means there are precisely n possibilities: 1, 3, .., 2n-1.
However, you have chosen n+1 numbers. So two of those numbers must have the same 's' part.
But then if the numbers were 2^a * s and 2^b * s, one of those numbers is a multiple of the other - assuming a>b, (2^a * s)/(2^b * s) = 2^(a-b) (and similarly for a<=b).