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Combinatorial design is a field of combinatorial mathematics that deals with arrangements of elements in sets that meet specific criteria. At the heart of this concept is the Latin square, which is a square array of size 'n' with 'n' different symbols, where each symbol occurs exactly once in each row and each column. Latin squares are used in various applications, such as experimental design, tournament scheduling, and error correcting codes.

Orthogonal Latin squares take this concept a step further by pairing two Latin squares such that each ordered pair of symbols occurs exactly once when the squares are superimposed. The exercise of finding two orthogonal Latin squares of order 7 is a practical example of constructing a combinatorial design, which requires not just arranging numbers in a pattern, but ensuring that all conditions for orthogonality are met. It is a classic example of how combinatorial design helps in solving problems where organization and pairing are key.

Abstract algebra is the study of algebraic structures such as groups, rings, and fields. In the context of orthogonal Latin squares and combinatorial design, the algebraic structure relevant to this discussion is the group. A group is a set equipped with an operation that combines any two elements to form a third element, while satisfying four conditions: closure, associativity, the presence of an identity element, and the existence of inverse elements.

The solution to the exercise involves creating arrays that follow a specific pattern, which aligns with the principles of group theory. The elements of Latin squares can relate to group elements, where the operation of 'shifting' resembles group operation. Understanding the properties of abstract algebraic structures is crucial in creating and recognizing patterns that adhere to strict rules, such as those found in orthogonal Latin squares.

Group theory is a branch of abstract algebra dealing with sets that have a binary operation satisfying particular rules. Groups feature heavily in the construction and understanding of combinatorial designs like Latin squares.

In relation to the provided exercise, group theory facilitates the understanding of how elements are combined. When creating a Latin square, the shifting process is akin to an operation within a cyclic group of order 7, where each shift is essentially an addition modulo 7, a common group operation. Recognizing the Latin square as a manifestation of a group allows us to utilize group-theoretic methods to create and analyze the structures, ensuring they meet the necessary criteria, such as orthogonality.

Modular arithmetic, sometimes referred to as 'clock arithmetic', is a system of arithmetic for integers, where numbers wrap around after reaching a certain value, called the modulus. It is akin to the 12-hour clock rolling over after reaching 12 o'clock. In the context of orthogonal Latin squares, modular arithmetic is used when we are constructing the second square to make sure that numbers stay within the set range of 0-6.

In the step by step solution, after adding the row and column numbers, the modulo 7 operation ensures that the resulting square remains within the proper bounds, exhibiting a key use of modular arithmetic. This tool is invaluable not only in abstract algebra but also in computer science, cryptography, and number theory. It streamlines complex problems into more manageable ones by working within a finite system of numbers.