Distinct primes are coprime

#coprime#primes#distinct

Statement

Show that distinct primes $p,q$ are coprime.

Hint

The only divisors of $p$ are $1$ and $p$ .

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Divisors. $\gcd(p,q)$ divides $p$ , so it is $1$ or $p$ .

If $p$ . Then $p\mid q$ , but $q$ is prime with $q\neq p$ , contradiction (as $p>1$ ).

Conclusion. $\gcd(p,q)=1$ .