Modular Arithmetic

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Chapter 9 Modular Arithmetic

Is \(30^{99}+61^{100}\) divisible by 31? This seems like a daunting question. However, in a couple of pages, you'll learn a trick that will make this and other problems involving divisibility of very large numbers much easier. This method involves modular arithmetic.

The framework of modular arithmetic as we are familiar with it was first developed by Gauss in his book Disquisitiones Arithmeticae in 1801, whereas many fundamental results that are now stated in terms of congruences were in fact proven much earlier; for example, Fermat's Little Theorem below was proven in 1640, though now it is standard to formulate it using congruences.

Despite its classical roots, modular arithmetic is important for both mathematics and applications. In mathematics, it is fundamental for studying abstract algebra, which you will learn more about in Fundamentals of Pure Math next year and in Honours Algebra the year after that. Moreover, it is also a cornerstone of modern cryptography.