Question

Prove that the fraction 7n+1/20n+1 is irreducible for every natural number n

Answer 1

Answer:

Suppose $\frac{7n+1}{20n +1} = k$

So

$7n+1 = k(20n +1)\\(7 + 20k) n = 1k - 1\\n = \frac{1k - 1}{7 + 20k}$

Since 1k - 1 < 7 + 20k, n is not a natural number

So, $\frac{7n+1}{20n +1}$ is irreducible for every natural number n.