If aN = {an : n ∈ N} and bN ∩ cN = dN, where a, b, c ∈ N and b, c are coprime, then

d = bc

We have bN = {bn : n ∈ N}, cN = {cn : n ∈ N} and dN = {dn : n ∈ N}

We have bN ∩ cN = dN

∴ d = d . 1 ∈ bN ∩ cN

⇒ d = bn₁ and d = cn₂ where n₁, n₂ ∈ N

⇒ b/d and c/d ⇒ bc/d, because b and c are coprime.

Also bc ∈ bN and bc = cb ∈ cN

∴ bc ∈ bN ∩ cN or bc ∈ dN

⇒ bc = dn₃ for n₃ ∈ N ⇒ d/bc

∴ bc = d

Hence (3) is the correct answer.