We consider positively existentially definable sets in the structure 〈ℤ; 1, +, ⊥〉 It is well known that the elementary theory of this structure is undecidable while the existential theory is decidable. We show that after the extension of the signature with the unary '-' functional symbol, binary symbols for dis-equality ≠ and GCD(·, ·) = d for every fixed positive integer d, every positive existential formula in this extended language is equivalent in ℤ to some positive quantifier-free formula. Then we get some corollaries from the main result. The binary order ≤ and dis-coprimeness ⊥ relations are not positively existentially definable in the structure 〈ℤ; 1, +, ⊥〉. Every positively existentially definable set in the structure 〈N; S, ⊥〉 is quantifier-free definable in 〈N; S, ≠, 0, ⊥〉. We also get a decidable fragment of the undecidable ∀∃-Theory of the structure 〈ℤ; 1, +, ≤, |〉, where | is a binary predicate symbol for the integer divisibility relation.