Help, I'm Trapped in a Math Factory!

Algebra Level 5

A certain factory produces numbers to be used in math problems. A worker there named Alex is given a batch of \(N\) numbers. First, he removes one number for inspection, then packages half of the remaining numbers. From the remaining numbers, he removes one and packages half the numbers after that. He repeats this process \(2014\) more times (remove one, package half the remaining), and at the end, he is left with one number that he keeps for himself. Find the units digit of \(N\).

Notes:

After the first sorting, Alex is left with \(\frac{1}{2}(N-1)\) numbers. After the second sorting, he is left with \(\frac { 1 }{ 2 } \left(\frac { 1 }2(N-1)-1\right)\) numbers, and so on.

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