Not sure where to go with this proof Prove or disprove for a

Not sure where to go with this proof:

\"Prove or disprove for an arbitrary prime number p there exists some composite number q where gcd(p, p+q) > 1\"

I know that if q is composite, it has a prime number that will evenly divide it such that this prime number is less than or equal to square root of q. I also know that q can be written as q = a * b, such that a divides q and b divides q. And I know that if p is a prime number, then p > 1 and can be written as p = p * 1.

But I\'m not sure how to relate these into a proof or a counter example.

Please assist.

Solution

let a prime number p be given

we know that 2p is composite and

(p,2p) = p >1

=>
for every prime number p , there exists a composite number q(take q=2p) where gcd(p,q) >1

thus proved