This paper investigates a surprising pattern that connects the divisibility properties of factorials to the primality of consecutive integers. Specifically, a certain ratio involving factorials and triangular numbers is shown to be an integer if and only if the consecutive integer is prime. A proof of this result is provided along with an alternative argument offering additional insight. Although not intended as a practical primality test, this characterization highlights intriguing connections between factorial properties and prime behavior, drawing parallels to classical results such as Wilson’s theorem.