Art of Problem Solving.pdf

# Art of Problem Solving.pdf - Art of Problem Solving Modular...

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2/17/2018 Art of Problem Solving 1/8 Modular arithmetic is a special type of arithmetic that involves only integers. This goal of this article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily solved using modular arithmetic. 1 Motivation 2 Residue 3 Congruence 3.1 Examples 3.2 Sample Problem 3.2.1 Solution: 3.2.2 Another Solution: 4 Making Computation Easier 4.1 Addition 4.1.1 Problem 4.1.2 Solution 4.1.3 Why we only need to use remainders 4.1.4 Solution using modular arithmetic 4.1.5 Addition rule 4.1.6 Proof of the addition rule 4.2 Subtraction 4.2.1 Problem 4.2.2 Solution 4.2.3 Subtraction rule 4.3 Multiplication 4.3.1 Problem 4.3.2 Solution 4.3.3 Solution using modular arithmetic 4.3.4 Multiplication rule 4.4 Exponentiation 4.4.1 Problem #1 4.4.2 Problem #2 4.4.3 Problem #3 5 Summary of Useful Facts 6 Applications of Modular Arithmetic 7 Resources 8 See also Let's use a clock as an example, except let's replace the at the top of the clock with a . Modular arithmetic/Introduction Contents Motivation

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2/17/2018 Art of Problem Solving 2/8 Starting at noon, the hour hand points in order to the following: This is the way in which we count in modulo 12 . When we add to , we arrive back at . The same is true in any other modulus (modular arithmetic system). In modulo , we count We can also count backwards in modulo 5. Any time we subtract 1 from 0, we get 4. So, the integers from to , when written in modulo 5, are where is the same as in modulo 5. Because all integers can be expressed as , , , , or in modulo 5, we give these integers their own name: the residue classes modulo 5. In general, for a natural number that is greater than 1, the modulo residues are the integers that are whole numbers less than : This just relates each integer to its remainder from the Division Theorem. While this may not seem all that useful at first, counting in this way can help us solve an enormous array of number theory problems much more easily! We say that is the modulo- residue of when , and . There is a mathematical way of saying that all of the integers are the same as one of the modulo 5 residues. For instance, we say that 7 and 2 are congruent modulo 5. We write this using the symbol : In other words, this means in base 5, these integers have the same residue modulo 5: Residue Congruence
2/17/2018 Art of Problem Solving

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