Not sure where to go with this proof: \"Prove or disprove for an arbitrary prime

Question

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.

Explanation / Answer

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

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