Question

Prove that if a and b are positive integers whose sum is a prime p, their greatest common divisor is 1.

Answer 1

Proof by contadiction.

Let assume there exist such positive integers $a$ and $b$ whose sum is a prime number $p$, that their greatest common divisor is greater than 1.

$a,b\in\mathbb{Z^+}\\d=\text{gcd}(a,b)>1\\\\a=de\\b=df\\e,f\in\mathbb{Z^+}\\\\a+b=p\\de+df=p\\d(e+f)=p$

Since $p$ is a prime number and $d>1$, then $d=p \wedge e+f=1$, but we assumed earlier that $e,f\in\mathbb{Z^+}$, and there are no two positive integers that sum up to 1.

q.e.d.