\(6\) Men and a Monkey

Algebra Level 5

\(6\) men are sharing a big pile of bananas with a monkey. The first gives one banana to the monkey and takes some bananas, leaving \(1/64\)th of the remaining pile. The second gives one banana to the monkey and takes some bananas, leaving \(1/32\)th of the remaining pile. This pattern continues, with the \(n\)th person leaving \(1/2^{7-n}\)th of the pile after giving a single banana to the monkey each time. Finally, the \(6\)th man gives one banana to the monkey, takes half of the pile of bananas, and leaves the rest for the monkey.

Let the smallest amount of total bananas needed such that each person receives a positive integer number of bananas be \(N\). What is \(N\pmod{1000}\)?

Note: The men do take some bananas for themselves.

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