Let the sequence $a\_n$ be defined as $a\_n = 2^n$ where $n = 0, 1, 2, \ldots $

That is, the sequence is $1, 2, 4, 8, \ldots$

Now assume I am told that a certain number is obtained by taking some of the numbers in the above sequence and adding them together (e.g. $a\_4 + a\_{19} + a\_5$), I may be able to work out what numbers from the sequence were used to arrive at the number. For example, if the number given is 8, I know that the sum was simply $a\_3$, since this is the only way to get 8. If i'm told the number is 12, I know that the sum was simply $a\_3 + a\_2$. 

If I'm told what the resulting number, can you prove that I can always determine which elements of the above sequence were used in the sum to arrive at that number?