A coprimality proof

#coprime#proof#linear combination

Statement

Show that for any integer $n$ , $2n+1$ and $3n+2$ are coprime.

Hint

Find a linear combination equal to $1$ .

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Combination. $3(2n+1)-2(3n+2)=-1$ .

Conclusion. So $2(3n+2)-3(2n+1)=1$ , giving a Bézout relation. Hence $\gcd(2n+1,3n+2)=1$ .

Application. The fraction $\frac{2n+1}{3n+2}$ is irreducible for all $n$ .