Modular arithmetic proofs


In mathematicsmodular arithmetic is a system of arithmetic for integerswhere numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticaepublished in A familiar use of modular arithmetic is in the hour clockin which the day is divided into two hour periods.

If the time is now, then 8 hours later it will be Because the hour number starts over after it reaches 12, this is arithmetic modulo In terms of the definition below, 15 is congruent to 3 modulo 12, so "" on a hour clock is displayed "" on a hour clock. Congruence modulo n is a congruence relationmeaning that it is an equivalence relation that is compatible with the operations of additionsubtractionand multiplication.

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Congruence modulo n is denoted:. The parentheses mean that mod n applies to the entire equation, not just to the right-hand side here b. This notation is not to be confused with the notation b mod n without parentheseswhich refers to the modulo operation. However, the b here need not be the remainder of the division of a by n.

That is. Subtracting these two expressions, we recover the previous relation:. Another way to express this is to say that both 38 and 14 have the same remainder 2—when divided by The congruence relation satisfies all the conditions of an equivalence relation :. However, the following is true:. The modular multiplicative inverse is defined by the following rules:. When the modulus n is known from the context, that residue may also be denoted [ a ]. Each residue class modulo n may be represented by any one of its members, although we usually represent each residue class by the smallest nonnegative integer which belongs to that class [2] since this is the proper remainder which results from division.

Any two members of different residue classes modulo n are incongruent modulo n. Furthermore, every integer belongs to one and only one residue class modulo n. Any set of n integers, no two of which are congruent modulo nis called a complete residue system modulo n. The least residue system is a complete residue system, and a complete residue system is simply a set containing precisely one representative of each residue class modulo n.

Some other complete residue systems modulo 4 include:. In fact, this inclusion is useful when discussing the characteristic of a ring.

Modular Arithmetic proofs (multiplication and addition mod n)

The ring of integers modulo n is a finite field if and only if n is prime this ensures that every nonzero element has a multiplicative inverse. In theoretical mathematics, modular arithmetic is one of the foundations of number theorytouching on almost every aspect of its study, and it is also used extensively in group theoryring theoryknot theoryand abstract algebra.

In applied mathematics, it is used in computer algebracryptographycomputer sciencechemistry and the visual and musical arts. A very practical application is to calculate checksums within serial number identifiers. In chemistry, the last digit of the CAS registry number a unique identifying number for each chemical compound is a check digitwhich is calculated by taking the last digit of the first two parts of the CAS registry number times 1, the previous digit times 2, the previous digit times 3 etc.

RSA and Diffie—Hellman use modular exponentiation. In computer algebra, modular arithmetic is commonly used to limit the size of integer coefficients in intermediate calculations and data. It is used in polynomial factorizationa problem for which all known efficient algorithms use modular arithmetic.Let a, b, and m be integers.

modular arithmetic proofs

Notation: means that a is congruent to b mod m. Note that if and only if. Thus, modular arithmetic gives you another way of dealing with divisibility relations. Congruence mod m is an equivalence relation :.

Sinceit follows that. Thenso. Suppose and. Then there are integers j and k such that. This implies that. Each integer belongs to exactly one of these classes. Two integers in a given class are congruent mod 3. If you know some group theory, you probably recognize this as constructing from.

For example,because as integers, and the congruence class of 3 is represented by 0. Likewise, as integers, and the congruence class of 4 is represented by 1.

I could have chosen different representatives for the classes say 3, -4, and 4. A choice of representatives, one from each class, is called a complete system of residues mod 3. But working mod 3 it's natural to use the numbers 0, 1, and 2 as representatives and in general, if I'm working mod n, the obvious choice of representatives is the set. This set is called the standard residue system mod nand it is the set of representatives I'll usually use. Thus, the standard residue system mod 6 is.

I'll prove the first congruence as an example. Then and for someso. There are no "fractions" mod I want to divide by 3, and to do this I need to multiply by the multiplicative inverse of 3. So I need a number k such that. A systematic way of finding such a number is to use the Extended Euclidean algorithm.

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In this case, I just use trial and error. Obviously, and won't work, so I'll start at :. Notation: or. Do not use fractions.Hot Threads. Featured Threads. Log in Register.

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Search titles only. Search Advanced search…. Log in. Contact us. Close Menu. JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding. Modular Arithmetic proofs multiplication and addition mod n. Thread starter Elwin. Martin Start date Sep 7, Homework Statement Let n be a fixed positive integer greater than 1. My reasoning is that if we treat mod n as an operation on a number, then we can mod n twice and we should get the same thing since any remainder divided by the same number should yield the same number.

I know that my work isn't very rigorous and I didn't really apply the definition I have directly, can anyone point me in another direction if this is the wrong approach? Thank you for your time, Elwin. SammyS Staff Emeritus. Science Advisor. Homework Helper. Gold Member. Martin said:.

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Is this a valid approach? Sorry if this is a bit silly to ask, but how does one show modular arithmetic operations rigorously in general? Thank you for your response, Elwin. I have to admit that I was thinking of something called 'congruence mod n', which is somewhat different, although closely related. You will probably cover it soon, if you haven't already done so. The 'mod' you have here appears to be a mathematical operation.

SammyS said:. Sorry, I've been traveling rhe last two days. Yes, it's true, but not important. Log in or register to reply now!This article collects together a variety of proofs of Fermat's little theoremwhich states that. Indeed, if the previous assertion holds for such amultiplying both sides by a yields the original form of the theorem. This is perhaps the simplest known proof, requiring the least mathematical background.

It is an attractive example of a combinatorial proof a proof that involves counting a collection of objects in two different ways. The proof given here is an adaptation of Golomb 's proof.

To keep things simple, let us assume that a is a positive integer. Consider all the possible strings of p symbols, using an alphabet with a different symbols. The total number of such strings is a psince there are a possibilities for each of p positions see rule of product. Let us think of each such string as representing a necklace. That is, we connect the two ends of the string together and regard two strings as the same necklace if we can rotate one string to obtain the second string; in this case we will say that the two strings are friends.

In our example, the following strings are all friends:. Notice that in the above list, each necklace with more than one symbol is represented by 5 different strings, and the number of necklaces represented by just one string is 2, i. One can use the following rule to work out how many friends a given string S has:. If we rotate it one symbol at a time, we obtain the following 3 strings:.

There aren't any others, because ABB is exactly 3 symbols long and cannot be broken down into further repeating strings. Using the above rule, we can complete the proof of Fermat's little theorem quite easily, as follows.

Our starting pool of a p strings may be split into two categories:. This proof uses some basic concepts from dynamical systems. For example, the function T 3 x is illustrated below:. Lemma 1. In other words, T mn x is the composition of T n x and T m x. For this point the lemma is clearly true, since. Now let us properly begin the proof of Fermat's little theorem, by studying the function T a p x.

From Lemma 1, we know that it has a p fixed points. By Lemma 2 we know that. We are interested in the fixed points of T a p x that are not fixed points of T a x. Let us call the set of such points S.

The main idea of the proof is now to split the set S up into its orbits under T a.

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What this means is that we pick a point x 0 in Sand repeatedly apply T a x to it, to obtain the sequence of points. This sequence is called the orbit of x 0 under T a.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I'd say you are right and the video is wrong. Further, the argument uses unidirectional inferences where bidirectional inferences are required. Below is one correct way to do the proof in that manner.

It's simpler to use basic rules of modular arithmetic. By the Congruence Product Rule we deduce. If someone in their comments indeed says that, then your doubts are well justified — that step is plain wrong. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Proof of modulo property Ask Question. Asked 1 year, 10 months ago. Active 1 year, 10 months ago. Viewed times.

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modular arithmetic proofs

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[Discrete Mathematics] Congruency Proof Examples

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modular arithmetic proofs

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modular arithmetic proofs

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