(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[itex]a[/itex] and [itex]b[/itex] are coprime. Show that for any [itex]n[/itex], there exists a nonzero integer [itex]k[/itex] that makes [itex]a+bk[/itex] and [itex]n[/itex] coprime.

2. Relevant equations

[itex]a[/itex] and [itex]b[/itex] are coprime if any of the following conditions are met:

- [itex]\text{gcd}(a,b)=1[/itex]
- the ideal [itex](a,b)=\{ax+by : x,y\in\mathbb{Z}\}[/itex] is equal to the set of all integers [itex]\mathbb{Z}[/itex]

3. The attempt at a solution

I tried expanding the desired result in terms of ideals:

[itex](a+bk,n) = \{(a+bk)x+ny : x,y\in\mathbb{Z}\} = \{ax+bkx+ny : x,y\in\mathbb{Z}\}[/itex]

If [itex]a[/itex] and [itex]n[/itex] are coprime, then setting [itex]k=0[/itex] makes [itex]a+bk[/itex] and [itex]n[/itex] coprime.

I couldn't figure out the case where [itex]a[/itex] and [itex]n[/itex] have a gcd other than 1.

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# Number theory: gcd(a,b)=1 => for any n, gcd(a+bk,n)=1 for some k

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