Methods to prove $\frac{21n + 4}{14n + 3}$ is irreducible for every natural number $n$

Question

Prove $\frac{21n + 4}{14n + 3}$ is irreducible for every natural number $n$.

I was thinking of taking a number-theory based approach.

Can you suggest the following method

Calculus/Number theory based methods? Please take a look at my attempt here.

Assume $\frac{21n + 4}{14n + 3}$ is reducible so we can apply modular arithmethic, considering the numerator and denominator seperately. Ideas here,

$21n + 4 \equiv 4 (\mod 7)$ $14n + 3 \equiv 3 (\mod 7)$

$35n + 7 \equiv 7 (\mod 7)$

Taking the LHS separately, $35n + 7 (\mod 7) \equiv DNE$ there is no residue since there is no remainder.

Therefore, by contradiction, it is true?

Thanks!

Answer 1

We want to show that $21n+4$ and $14n+3$ are relatively prime. Note that $$(3)(14n+3)-(2)(21n+4)=1.$$ So any common divisor of $21n+4$ and $14n+3$ divides $1$.

Remark: I am not able to construct an argument based on the post. The remainder when $(21n+4)+(14n+3)$ is divided by $7$ is $0$, I would not use DNE, since $0$ is a perfectly respectable remainder. But working modulo $7$ is not enough, we want to rule out all common divisors greater than $1$.

Answer 2

gcd(21n+4, 14n+3) = gcd(14n+3, 7n+1) = gcd(7n+2, 7n+1) = gcd(7n+1, 1) = 1

Using repeated usage of Euclidean algorithm.

Answer 3

A simple one-line proof: $$\gcd(21n+4,14n+3)=\gcd(7n+1,14n+3)=\gcd(7n+1,7n+2)=1.$$