Modular arithmetic problems Modular (often also Modulo) Arithmetic is an unusually versatile tool discovered by K. Find the last digit of 7100 3. Modular arithmetic involves performing operations within a specific modulus. [Solution: 6] 2. What is the invrese of 7 mod 48 (if it exists). 1. ) Find the greatest integer multiple of 8, no two of whose digits are Arithmetic Modulo n Useful Facts. What is modular arithmetic systems, and play an important role both in theoretical and applied mathematics. Therefore, 38 Time-keeping on this clock uses arithmetic modulo 12. Applications of modular arithmetic Hashing, pseudo-random numbers, ciphers. In particular, they occur in calendrical and astronomical problems since these involve cycles (man-made or natural), but one also finds modular arithmetic in purely mathematical problems. The Fundamental Theorem of Arithmetic says that every positive integer greater than 1 has a unique prime factorization, up to reordering (i. Solved word math problems, tests, exercises, and preparation for exams. Reduce the following numbers using modular arithmetic: Apr 17, 2022 · The term modular arithmetic is used to refer to the operations of addition and One of the basic problems dealt with in modern algebra is to determine if the Mar 9, 2024 · The probability can be a huge number, and you are often asked to output it modulo 10^9 + 7. Lecture 11 2 Modular Arithmetic Warm-Up! Try these problems before watching the lesson. It is currently 4:30 p. This lesson will cover numerous goals, such as Modular arithmetic is a system of arithmetic for integers, which considers the remainder. Now, let’s dive into some practical problem-solving scenarios where modular arithmetic can be applied effectively. Randolph High School Math League 2014-2015 Page 1 Introduction to Modular Arithmetic 1 Introduction Modular arithmetic is a topic residing under Number Theory, which roughly speaking is the study of integers and their properties. The sneaky thing about modular math is we’ve already been using it for keeping time — sometimes called “clock arithmetic”. Practice Problems on Sep 22, 2013 · Modular arithmetic GCD Divisibility rules Competition problems Problem (2003 AIME II, Problem 2. Problem 7. Problem 1: The Josephus Problem tions in various problems, including subsets of integers. modular arithmetic systems, and play an important role both in theoretical and applied mathematics. disher Solution 2 (Powers of 9) We need to first convert into a regular base-number: Modular arithmetic is a way of systematically ignoring differences involving a multiple of an integer. Part II contains 4 multi-step word problems involving modular arithmetic, with solutions provided for each. Many people prefer to write “a≡ b (mod m)”. Thus the last digit is a 2 when n is 1, 5, 9, etc. IUSACO: 13. Thus, modular arithmetic gives you another way of dealing with divisibility relations. xis congruent to ymod nif n| x−y. 2 The standard representatives for all possible numbers modulo 10 are given by 0,1,2,3,4,5,6,7,8,9 although, for example, 3 ≡ 13 ≡ 23(mod 10), we would take the smallest positive such number which is 3. Next, we know Modular arithmetic basics Review of . from the rst set of Practice Problems. It is not difficult to see that if p is prime then gcd(b,p) = 1 for all b less than p, therefore we can say that in mod p, every number except 0 has an inverse. Then, the required expression for the number of hot dog packages we should buy is 8 n mod 8 Use the rules of modular arithmetic to solve the following problems. e. The book includes video lectures for every chapter, formulas for every topic, and 3. Level 1 Exploring Numerical Patterns. Reviewing arithmetic modulo 12 We have previously worked through a modular arithmetic handout in which we focused much of our efforts on constructing the modular numbers from the integers. For example: it’s 7:00 (am/pm doesn’t matter). Modular arithmetic in Solving Linear Congruence: Modular arithmetic can be used to solve linear congruence, which are equations of the form ax ≡ b (mod m). The base-nine representation of the number is What is the remainder when is divided by . Actually, the modulo is there to make the calculation easier, not harder. Problems count 25 Similarly to how we used 12 and 60 as a modulus for modular arithmetic, any positive integer can be used. If a biology experiment begins at 7:00 AM and runs for 80 hours, at what time will it end? 2. Sep 13, 2015 · Modular Arithmetic Practice Joseph Zoller September 13, 2015 Practice Problem Solutions 1. Problem 1. Problem. Example. Authors: Darren Yao, Michael Cao, introduces modular arithmetic through numerous math-contest-level examples and problems. Problem 9. For example, in classic arithmetic, adding a positive number a to another number b always produces a number larger than b. sty code. (Pick an appropriate modulus for each. Notation: x= y (mod n). Modular arithmetic properties Congruence, addition, multiplication, proofs. What is k? 4. This throws away some of the information, but is useful because there are only finitely many remainders to consider. For examples, see Introduction to modular 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. (1992 AHSME 17) The two-digit integers form 19 to 92 are written consecutively to form the large integer N = 192021 909192. Consider the theorem: Let a;b 2Z and let d be the largest integer dividing both a and b (we call d the greatest common divisor of a and b, and we write d = gcd(a;b)). Two numbers are congruent modulo n if their difference is a multiple of n. Solving problems on modular arithmetic Problem 1 Find the number of solutions to this system of three modular equations N = {2 mod 5}, N = {2 mod 6}, N = {2 mod 7}. 2019 03:38 am Chapter: 10th Mathematics : UNIT 2 : Numbers and Sequences Feb 25, 2025 · 5. Do not attempt these problems without a solid foundation in the subject and use them for exam practice. Equivalence relations. Beck and R. This gives us 33 hours with a remainder of 42 minutes. com Problem Questions with Answer, Solution | Mathematics - Exercise 2. Level 3. Remark. An Introduction to Modular Arithmetic Brandon Jiang May 15, 2016 1 Introduction Modular arithmetic is an important topic in number theory dealing largely with the study of remainders. (a) Consider congruence mod 3. Using these concepts, we also define the Euclidean algorithm, modular arithmetic, and primitive roots. Textbook: We are using "The Art of Proof: Basic Training for Deeper Mathematics" by M. Modular Arithmetic Word Problems. For ,, with 0 ≡ (mod )↔ | ( −) Definition: “a is congruent to b modulo m” Integers Domain of Discourse This is a predicate (T/F values) on integers ,, . The document encourages the user to try more questions from their textbook and past exams to further practice In some sense, modular arithmetic is easier than integer arithmetic because there are only finitely many elements, so to find a solution to a problem you can always try every possbility. What time will it be in 2022 minutes? We know that every 60 minutes will be an hour, so we divide 2022 by 60 to figure out how many hours will pass. iff their difference is exactly divisible by N. Prove, using modular arithmetic, the following divisibility statements. Find the last digit of 7100 [Solution: 1] 7100 (72) 50 49 ( 1)50 1 mod 10. Problem: If 17 hours have passed since 3:00 PM, what time is it now in 12 Modular Arithmetic In studying the integers we have seen that is useful to write a = qb+r. Instead of waiting until we obtain the final answer before we reduce it modulo \(n\), it is easier to reduce every immediate result modulo \(n\) before moving on to the next step in the computation. Modular arithmetic highlights the power of remainders when solving problems. Problem-Solving with Modular Arithmetic. Art of Problem Solving Modular Arithmetic. Show that 13|6123123 −8 3. Lecture 11 2 Note that a = 0 (mod m) if and only if m | a. So it must be 2. What is modular arithmetic? Modulo operator. Suppose that 3k is the highest power of 3 that is a factor of N. Modular arithmetic and integer representations Unsigned, sign-magnitude, and two’s complement representation. m. Usually (and on this page) a,b, are nonnegative and N a positive integer. (a) What are the equivalence classes under the relation of congruence mod 3? (b) Construct an addition table for addition mod 3. Let n, x, and ybe integers. Both 38 and 14 have the same remainder (2) when divided by 12. Addition. What is n mod 8? In the rst problem, 34 mod 8 is 2. Modular arithmetic motivates many questions that don’t arise when study-ing classic arithmetic. Division: , where is a positive integer that divides and . There are n residue classes modulo n. Given that 5x 6 (mod 8), nd x. 5. 05. Subtraction: . Modular Arithmetic Problem (2015 PUMaC NT A1) What is the 22nd positive integer n such that 22n ends in a 2? (when written in base 10) Solution: The powers of 22, modulo 10, are 2, 4, 8, 6, 2, . Modular Arithmetic Warm-Up! Try these problems before watching the lesson. Solution Consider integer number N-2. Another way of saying this is: Mod many multiple of mis 0. Two numbers a and b are said to be equal or congruent modulo N iff N|(a-b), i. In addition, all problems in the handout were either copied from the Art of Problem Solving Wiki or made by myself. In this lecture, I Ex. Find the remainders when 70004 and 778 is divided by 7. Now that you know Fermat’s Little Theorem, solve Problem 3 in a (mildly) simpler way than before. 75 Lessons. Digital SAT Math Problems and Solutions (Part - 172) Read More. The 22nd term in this sequence is 85 . Time-keeping on this clock uses arithmetic modulo 12. F. Multiplication: . Exponentiation: where is a positive integer. May 24, 2024 · While solving modular arithmetic problems, we directly operate on the remainder without tedious computations, following the given rules or properties. We say a b (mod n) if n divides a b. (a)3jk3 k, for all k 2N, MODULAR ARITHMETIC WORKSHEET. These types of problems often appear in number theory and cryptography. Gauss (1777-1855) in 1801. If A + B = C, then A (mod n) + B (mod n) ≡ C (mod n) If A ≡ B (mod n), then A + k ≡ B + k (mod n) for any integer ‘k’ See full list on calcworkshop. Modular arithmetic is often tied to prime numbers, for instance, in Wilson's theorem, Lucas's theorem, and Hensel's lemma, and Modular arithmetic is a generalization of parity. This may sound counterintuitive, but once you know how modular arithmetic works, you'll see why too. It does not produce numbers as output. omegalearn. Modular arithmetic modulo \(n\) uses the mod operation to reduce the answers of all computation to within 0 through \(n-1\). Solved Examples of Modular Arithmetic. 3: Modular Arithmetic | 10th Mathematics : UNIT 2 : Numbers and Sequences Posted On : 28. Exercises 1. Oct 30, 2022 · Get Mastering AMC 10/12 book: https://www. Part I contains 11 questions asking users to find results of expressions modulo given numbers. Clock Math. Show that there are no integer solutions to w6 + x6 + y6 + z6 = 1012 + 7 or w6 + x6 + y6 + z6 = 1012 + 4. We now have a good definition for division: \(x\) divided by \(y\) is \(x\) multiplied by \(y^{-1}\) if the inverse of \(y\) exists, otherwise the answer is Problem. Rather than giving an account of properties of modular arithmetic, we give examples of its applications to contests. Be clear which n are you choosing when you perform computations modulo n. Important! These are all exam-level problems. That is every integer is congruent to one of 0;1;2;3;:::;n 1 modulo n. Aug 20, 2017 · In the Modular Arithmetic section, we stated a theorem that says if gcd(a,m) = 1 then a-1 (the inverse of a) exists in mod m. Since equality mod mis an equivalence relation, For further information on this math subject, make sure to view the complementary lesson named Modular Arithmetic: Examples & Practice Problems. What digits can an integer square end in? State and prove a theorem. Modular arithmetic is a special type of arithmetic that involves only integers. Solving equations in modular arithmetic involves setting two sides equal and finding the missing number that satisfies the Modular Arithmetic. Math questions with answers and solved math homework. Consider four integers and a positive integer such that and . Assume it is 7:00, and we want to know what time it will be 10 Modulo - math problems. What day of the week will his birthday be on in 2016? 3. Level 5 mathematically model SET are the modular arithmetic fields. 3 of an integer (use arithmetic modulo 13). Level 2. In Modular arithmetic problems + functions problems 1. Explore Modular Arithmetic by working with remainders to solve problems about very large numbers. Where will the hour hand be in 7 hours? Hrm. Modulo Challenge. Clock Arithmetic: The concept of modular arithmetic is akin to how clocks work, where the hours wrap around after reaching 12 or 24. In this lecture, I Modular Arithmetic Practice Problems. Modular arithmetic May 23, 2025 · Examples of the use of modular arithmetic occur in ancient Chinese, Indian, and Islamic cultures. In Modular arithmetic is a system where numbers reset to zero upon reaching the modulus. ) Problem 8. 12 = 22 23 = 3 2 ). . In modular arithmetic, numbers "wrap around" upon reaching a given fixed quantity (this given quantity is known as the modulus) to leave a remainder. It includes the following topics: Modular arithmetic; Modular exponentiation; Greatest Common Divisor (GCD) Extended Euclidean algorithm; Modular multiplicative inverse; 1. 2. This article discusses topics that are frequently used to solve programming problems based on math. Let us first discuss what it means to say that a ≡ b mod m, that is, “a is congruent to b modulo m When dealing with time-based calculations, such as adding hours to a given time, modular arithmetic (usually modulo 24 or 12) is often employed. We had to purchase 4 packages of hot dog buns so that this remainder would not occur, as in (34 mod 8) 4 = 2 4 = 8, which is divisible by 8. Show that Dec 29, 2023 · Intro to Modular Arithmetic The Math of Time. freeman66 (May 13, 2020) Modular Arithmetic in the AMC and AIME §0Acknowledgements This was made for the Art of Problem Solving Community out there! I would like to thank Evan Chen for his evan. In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The study of the properties of the system of Modular Arithmetic (General Problems) Los Angeles Math Circle 4 July 2021 1. It satisfies to these three modular equations N-2 = {0 mod 5}, N-2 = {0 mod 6}, N-2 = {0 mod 7}. Soon you'll be solving these problems like second nature. Often we can solve problems by considering only the remainder, r. Congruence modulo. Sep 13, 2015 · Modular Arithmetic Practice Joseph Zoller September 13, 2015 Problems 1. If nis an integer, two integers are equal mod nif they differ by a multiple of n; it is as if multiples of nare “set equal to 0”. Nov 12, 2023 · These will be problems, proofs and such to either challenge the reader, make sure they understand or for a little bit of fun! Although, after each part, the solution will be provided. Terminology and notation Randolph High School Math League 2014-2015 Page 1 Introduction to Modular Arithmetic 1 Introduction Modular arithmetic is a topic residing under Number Theory, which roughly speaking is the study of integers and their properties. org/mastering-amc1012. There are opportunities for noticing patterns and discovering some general rules which hold in this type of arithmetic. See also: 2011 NT A3, 2013 NT A2, 2014 NT A2, 2015 NT B1 The problems in this feature introduce the idea of modular (or clock) arithmetic, and encourage you to explore the modular world. Example 2 This document contains two parts of a modular arithmetic practice problem set. Since modular arithmetic is such a broadly useful tool in number theory, we divide its explanations into several levels: Introduction to modular arithmetic; Intermediate modular arithmetic; Olympiad modular arithmetic Modular Arithmetic New notion of “sameness” or “equivalence” that will help us understand modular arithmetic. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. Level 4. So let us begin our journey into modular arithmetic. Level 1. In the Chapter on Set Theory, we discussed the idea of relations 1 and specifically provided the example of modular equivalence on this page. In this handout, we simply review how to do computations with these numbers. 3. ) As in our initial clock example, let's work in modulus 12. (3) Explore the powers of divisibility, modular arithmetic, and infinity. In modular arithmetic, the following identities hold: Addition: . Definition. Example 1: Problem: Show that 38 ≡ 14 (mod 12) Solution: 38 = 3 × 12 + 2 14 = 1 × 12 + 2. Moreover, we can define operations of addition and multiplication in the modular arithmetic: • To add two numbers in modular arithmetic, add them in the ordinary sense and then reduce (if necessary) in modular arithmetic; Modular arithmetic is a special type of arithmetic that involves only integers. This approach is extremely useful in a variety of difficult problems. Inverses in Modular arithmetic We have the following rules for modular arithmetic: Sum rule: IF a ≡ b(mod m) THEN a+c ≡ b+c(mod m). Geoghegan. It uses notation like "a ≡ b (mod n)" to indicate that a and b have the same remainder when divided by n. Here are examples of word problems involving addition, subtraction, multiplication, and division in modular arithmetic, along with their solutions: Addition. 7 + 7 = 14, but we can’t show “14:00” on a clock. 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. Congruence relation. Solution 1 (Modular Arithmetic) Recall that We expand by the definition of bases: ~Aidensharp ~Kante314 ~MRENTHUSIASM ~anabel. Cory’s birthday lies on a MOnday this year. 1Introduction Divisibility, prime numbers, and congruences are some of the most significant topics within number theory. The problems immediately above can be reduced to the following. rsxeb nbhecx hhsua anl srnwuga svcc wzd gxvebv rmjrwxvz qehzvcf