013. Irreducible Fraction

Prove that the fraction $\cfrac{21n+4}{14n+3}$ is irreducible for every natural number $n$.

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Assume that $(21n+4) / (14n+3)$ is a reducible fraction. $$\begin{aligned} \cfrac{21n+4}{14n+3} = 1 + \cfrac{7n+1}{14n+3} \end{aligned}$$

Then, $(7n+1) / (14n+3)$ must be a reducible fraction as well. So, its reciprocal is also reducible. $$\begin{aligned} \cfrac{14n+3}{7n+1} = 2 + \cfrac{1}{7n+1} \end{aligned}$$

However, since $1$ is coprime with $7n+1$ for every natural number $n$, this contradicts the assumption that $(21n+4) / (14n+3)$ is a reducible fraction. Therefore, $(21n+4) / (14n+3)$ is irreducible for every natural number $n$.

Reference

[1] 1959 IMO Problems/Problem 1