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.