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In this chapter, we will study modular arithmetic, that is, the arithmetic of congruence classes, where we simplify number-theoretic problems by replacing each integer with its remainder when divided by some fixed positive integer n. This has the effect of replacing the infinite number system ℤ with a number system ℤn which contains only n elements. We find that we can add, subtract and multiply the elements of ℤn (just as in ℤ), though there are some difficulties with division. Thus ℤn inherits many of the properties of ℤ, but being finite it is often easier to work with. After a thorough study of linear congruences (the analogues in ℤn of the equation ax = b), we will consider simultaneous linear congruences, where the Chinese Remainder Theorem and its generalisations play a major role.
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