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

This preview shows page 1 - 4 out of 8 pages.

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
Image of page 1

Subscribe to view the full document.

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
Image of page 2
2/17/2018 Art of Problem Solving
Image of page 3

Subscribe to view the full document.

Image of page 4

{[ snackBarMessage ]}

Get FREE access by uploading your study materials

Upload your study materials now and get free access to over 25 million documents.

Upload now for FREE access Or pay now for instant access
Christopher Reinemann
"Before using Course Hero my grade was at 78%. By the end of the semester my grade was at 90%. I could not have done it without all the class material I found."
— Christopher R., University of Rhode Island '15, Course Hero Intern

Ask a question for free

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern