If youre behind a web filter, please make sure that the domains. Modular arithmetic is also called clock arithmetic. Math circle thursday january 22, 2015 what is modular arithmetic. We say that a divides b or a is a divisor of b, denoted ajb, if there is a number c 2z such that b ac. Everything you need to know about modular arithmetic. In fact, this is the example we shall need for the rsa cryptosystem. Modular arithmetic is a special type of arithmetic that involves only integers. Modular arithmetic is a system of arithmetic for integers, which considers the remainder.
This printable worksheet and interactive quiz will provide an assessment of your proficiency at solving math problems using modular arithmetic. When can an integer be written as the sum of two squares. Integer and modular arithmetic jim carlson february 2, 2004 contents 1 introduction 3. Modular arithmetic summary this lesson is an exploration of modular arithmetic. Answer the following questions, but dont worry about ampm. The classical example for mod arithmetic is clock arithmetic. For example, in classic arithmetic, adding a positive number a to another number b always produces a number larger than b. There are many places where a minor trick is required, but the students may not always think of doing that for solving their problems. Your instructor will also provide information on accessing answerssolutions for these problems. Since such equations have many solutions we introduce the notation amodm. In modular arithmetic, we select an integer, n, to be our \modulus. In the 1700s, swiss mathematician leonard euler pioneered the modern conception of modular arithmetic. The only difference between modular arithmetic and the arithmetic you learned in your primary school is that in modular arithmetic all operations are performed regarding a positive integer, i.
This relation gives rise to an algebraic structure called the integers modulo usually known as the integers mod, or for short. Modular arithmetic was also considered in purely mathematical contexts, such as in sun zis mathematical manual. This set is called the standard residue system mod m, and answers to modular arithmetic problems will usually be simplified to a number in this range. The best way to introduce modular arithmetic is to think of the face of a clock. This means any integer is equivalent to the remainder of that number divided by 5. But when n is a prime number, then modular arithmetic keeps many of the nice properties we are used to with whole numbers. You must express your final answer as n, where \0 \le n \le 10\. Answer key modular arithmetic modular arithmetic uses the concept of remainder to relate numbers. Math challenge washington middle school march 4, 2010 warmup activity pretend that its 3. Here, the modulus is 12 with the twelve remainders 0,1,2,11. Modular arithmetic motivates many questions that dont arise when studying classic arithmetic. Learn more math and science with, first 200 people to sign up will get 20% off your subscription, and y.
Arithmetic practice questions solve the following problems and select your answer from the choices given. You may use the paper you have been given for scratch paper. Richard earl mathematical institute university of oxford. Modular arithmetic is often tied to prime numbers, for instance, in wilsons theorem, lucass theorem, and hensels lemma, and generally appears in fields. Modular arithmetic peter mcnamra bucknell university and trinity college dublin.
So repeated multiplication, that is taking powers is also possible. Modular arithmetic victor adamchik fall of 2005 plan 1. Assess your learning the last part of each lesson is a short assessment. Modular arithmetic especially the properties of congruence are an important tool in arriving at quick solutions to a variety of problems. In modular arithmetic, numbers wrap around upon reaching a given fixed quantity this given quantity is known as the modulus to leave a remainder. Weshow how todeal withmodular arithmeticoperations and inequalities for both linear and nonlinear problems. Read and learn for free about the following article. This is a \partition of the integers into two \classes. If you are using this material as part of a formal class, your instructor will provide guidance on which problems to complete. Olympiad number theory through challenging problems justin stevens third edition.
Both procedures are suitable for integration with nelsonoppen framework 1, 2, 3. Most technological applications of modular arithmetic involve exponentials with very large numbers. For example, a typical problem related to encryption might. As in our initial clock example, lets work in modulus 12. The students should get an opportunity to present solutions often if possible and be guided as they present, so that the. So we need some other way in which the problems can be solved in a very short time.
The sequence also continues into what would be the negative numbers. In multiplication tables modulo n, rows corresponding to numbers coprime with n contain permutations of the first row. The division algorithm also works in qx, the set of polynomials with rational. Show that an integer is divisible by 9 if and only if the sum of its digits is. Arithmetic practice questions harford community college. Modular arithmetic word problems on brilliant, the largest community of math and science problem solvers. The students should get an opportunity to present solutions often if possible and be guided as they present, so that the important ideas are highlighted to the. The value of a mod b is the remainder when you divide a by b. On clocks we start at 1 instead of 0, continue to 12, and then start at 1 again. Congruence modulo if youre seeing this message, it means were having trouble loading external resources on our website. So, when you give the time you actually give a remainder between 0 and 11. Modular arithmetic word problems practice problems. 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. Given integers, and, with, we say that is congruent to modulo, or mod, if the difference is divisible by for a given positive integer, the relation mod is an equivalence relation on the set of integers.
If none of them work then we know \y\ does not have an inverse. Modulo a prime number we have seen that modular arithmetic can both be easier than normal arithmetic in how powers behave, and more di. Olympiad number theory through challenging problems. In contrast to subjects such as arithmetic and geometry, which proved useful in everyday problems in commerce and architecture, as. In modular arithmetic, we are going to learn solving word problems in the following three sub topics. It starts with some occurrences of patterns in powers of numbers and in clock arithmetic. You must show your work, keeping all numbers in your calculations small. High school mathematics extensionsprimesmodular arithmetic. Arithmetic problems the best essay writing service. Now that you know fermats little theorem, solve problem 3 in a mildly simpler way than before.
The way we need to solve the problem quickly is called as shortcut. You will get even more practice on a quiz when youve finished the lesson. In some sense, modular arithmetic is easier than integer artihmetic because there are only finitely many elements, so to find a solution to a problem you can always try every possbility. Clearly from the condition on the set, we must have a bq. The modulo arithmetic lesson aims to teach students addition and multiplication modulo n. Modular arithmetic is also called clockarithmetic because we only use 12 numbers to tell standard time. The modular arithmetic properties allow me to solve this equation the way i would solve a linear equation, up to a point. In fact, in many ways taking powers is easier in modular arithmetic than in standard arithmetic. Modular arithmetic, cryptography, and randomness for hundreds of years, number theory was among the least practical of mathematical disciplines. Modular arithmetic practice joseph zoller september, 2015 practice problem solutions 1. Euclidean and division algorithm 6 by the wellordering principle we know that this set must have a minimum, say when q q 1.
In modular arithmetic, the numbers we are dealing with are just integers and the operations used are addition, subtraction, multiplication and division. Modular arithmetic, or clock arithmetic, reduces all positive integers to a set of remainders. There is, in addition, a section of miscellaneous problems. Buy the hardcover book arithmetic problems by e wentworth at, canadas largest bookstore. Practice using the modulo operator if youre seeing this message, it means were having trouble loading external resources on our website. I multiply out the left side, then get the xs on one side. Solving modular arithmetic questions mathematics stack.
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