To meet the company's requirements for private traffic between application servers and the S3 bucket without using public IP addresses, the most cost-effective solution is to use an S3 gateway endpoint. This is because gateway endpoints are free of charge and provide a secure and private connection between your VPC and S3 without requiring an internet gateway, NAT device, VPN connection, or AWS Direct Connect connection. By configuring an S3 gateway endpoint and modifying the route table with the appropriate route for the endpoint, the EC2 instances can access the S3 bucket through the gateway endpoint, ensuring that the traffic remains within the AWS network and does not traverse the public internet. This solution does not require any additional infrastructure or complex configurations, making it the most cost-effective option. Option A is the correct choice because it directly addresses the requirement for private access to S3 from the EC2 instances in the VPC without incurring additional costs.