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[(1 + 2^p + 2^{n-p})N - 1] / 2^n

Source: IMO ShortList 1990, Problem 21 (ROM 1)

August 15, 2008

number theoryDivisibilitybinary representationminimizationIMO Shortlist

Problem Statement

Let

n

be a composite natural number and

p

a proper divisor of

n.

Find the binary representation of the smallest natural number

N

such that \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n} is an integer.