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Modular Arithmetic
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Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations of the ring of integers: addition, subtraction, and multiplication. For a fixed modulus n, it is defined as follows.
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This observation underpins modular arithmetic. Sometimes such b is called the residue of a (mod n). If b is non-negative and smaller than |n| (the absolute value of n), then b is called common residue. This terminology has little in common with residues in complex analysis. The quantity n is sometimes called the base.
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The following discussions and activities are designed to lead the students to practice their basic arithmetic skills by learning about clock arithmetic (aka modular arithmetic) and cryptography. The lesson can be done individually or in groups of any size. It is long, taking approximately 2 hours, but can be separated easily into two lessons.
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[Due to problems that students in the class had with modular arithmetic in the class, here is a primer. Carefully note this primer not just for the results and examples but ... the format in which arguments and proofs are presented. This is what is expected in your homeworks and exams.]
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Students will be introduced to modular arithmetic by first examining a five-hour analog clock and its mathematical properties. Then students will investigate patterns and relationships that exist in 12-hour addition and multiplication clock tables.
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Since modular arithmetic has such a wide range of applications, it is important to know how hard it is to solve a system of congruences. A linear system of congruences can be solved in polynomial time with a form of Gaussian elimination, for details see the linear congruence theorem.
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Modular Arithmetic