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Sum of finite and infinite set

Source: China TST 2019 Test 2 Day 2 Q4

March 11, 2019
modular arithmeticnumber theoryChina TST

Problem Statement

Does there exist a finite set AA of positive integers of at least two elements and an infinite set BB of positive integers, such that any two distinct elements in A+BA+B are coprime, and for any coprime positive integers m,nm,n, there exists an element xx in A+BA+B satisfying x≡n(modm)x\equiv n \pmod m ?
Here A+B={a+b∣a∈A,b∈B}A+B=\{a+b|a\in A, b\in B\}.
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