Divisibility tests for digits other than 7 are well known and rely on the base 10 representation of numbers. For example, a natural number is divisible by 4 if the last 2 digits are divisible by 4 because 4 divides 10[sup k] for all k equal to or greater than 2. Divisibility tests for 7, while not nearly as well known, do exist and are also derived from the base 10 expansion. The aim of this paper is to provide an overview of known divisibility tests for 7, compare and contrast them; and then introduce a new algorithm discovered by Joshua Arzt while a student at Alfred University. The Arzt Algorithm will then be generalized based on techniques from existing tests. All of these tests provide excellent applications of modular arithmetic, a subject that appears in abstract algebra, number theory, foundations of mathematics, and problem-solving courses. This material is accessible enough for students to explore on their own, make conjectures, attempt proofs and give presentations. Writing programs for these divisibility tests is appropriate for many introductory computer science courses, and can lead to further discussions on time estimates for doing arithmetic. In mathematics education classes, where a deep understanding of base 10 representation as well as other bases is a focus, these tests make very nice group projects.