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Find a subset of Z+ satisfying a property

Source: Turkish TST 2012 Problem 9

March 26, 2012

modular arithmeticnumber theoryprime numbersnumber theory proposed

Problem Statement

Let

\mathbb{Z^+}

and

\mathbb{P}

denote the set of positive integers and the set of prime numbers, respectively. A set

A

is called

S-\text{proper}

where

A, S \subset \mathbb{Z^+}

if there exists a positive integer

N

such that for all

a \in A

and for all

0 \leq b <a

there exist

s_1, s_2, \ldots, s_n \in S

satisfying

b \equiv s_1+s_2+\cdots+s_n \pmod a

and

1 \leq n \leq N.

Find a subset

S

of

\mathbb{Z^+}

for which

\mathbb{P}

is

S-\text{proper}

but

\mathbb{Z^+}

is not.

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