Download Modulo Arithmetic: Solving Problems and Understanding Concepts and more Summaries Mathematical logic in PDF only on Docsity! Name: Coleen B. Nerves Course & Block: BSEd Math 3A Instructor: Edward B. Pescuela Date: February 14, 2022 MathEd 17: ABSTRACT ALGEBRA MODULE 1 MODULO ARITHMETIC Solve the following: Find the remainder when 123 + 234+ 32+ 56+ 22 + 12 + 78 is divided by 3. Find the last three digits of 240. 240 ≡¿ ≡ (24 ) 4 (mod 1000) ≡ (24 ) 3 ∙24 (mod 1000) ≡ 824 ∙24 (mod 1000) ≡19776 (mod 1000) 240 ≡776 (mod 1000) IV. Evaluation Solve the following: 1. Find the modulo 4 residue of 311. 311 (mod 4) 311 ÷ 4 = 77 r. 3 311 ≡ 3 (mod 4) So, 3 is the modulo 4 residue of 311. 2. What is the last digit of (. . . (7¿¿ 7 ¿ 7). . . ¿7if there are 1000 7s as exponents and only one 7 in the middle? This can also be stated as . After that, we see that 7 is congruent to -1 in mod 4, so we can use this fact to replace the 7s with -1s, because 7 has a pattern of repetitive period 4 for the units digit. is simply 1, so therefore , which really is the last digit. 0 + 0 + 2 + 2 + 1 + 0 + 0 = 5 (mod 3) = 5 (mod 3) 5 = 2 (mod 3) 123 ≡ 0 22 ≡ 1 234 ≡ 0 12 ≡ 0 32 ≡ 2 78 ≡ 0 56 ≡ 2 3. Jerry has 44 boxes of soda in his truck. The cans of soda in each box are packed oddly so that there are 113 cans of soda in each box. Jerry plans to pack the sodas into cases of 12 cans to sell. After making as many complete cases as possible, how many sodas will Jerry have leftover? Problem: Find the remainder when the product 44 ∙ 113 is divided by 12. or 44 ∙ 113 ≡ 8 ∙ 5 ≡ 40 ≡ 4(mod 12) Thus, Jerry has 4 sodas leftover after making as many cases of 12 as possible. V. Reflection 1. Write three (3) things you learn about the given concept. What interests me the most in this module is how modular arithmetic circulates between integers and their remainders (where the modulus is another name for the remainder after division). In general, I’ve learned that; a. Modular arithmetic works like clock arithmetic because it functions on a modulus of 12 as the latter one wrap around times past 12. b. For a positive integer n, two integers a and b are said to be congruent modulo n; if a and b have the same remainder when divided by n. c. To satisfies all the conditions of an equivalence relation, there are properties involving: Addition Multiplication Exponentiation Division 2. What difficulty did you encounter in the topics presented? I find it hard solving in the part of exponentiation and it is still confusing for me. 3. Write at least two questions that you can generate from the discussion. a. How do you solve modular arithmetic? b. How do you know if a modulo is congruent? VI. Assignment 1. Illustrate the concept of function. Function is an expression, rule, or law that defines a relationship between the independent and dependent variable. We use functions to describe how are variable changes as a consequence of another variable changing. In short, it can be thought of as a rule which operates on an input and produces an output. 44 (mod 12) ∙ 113 (mod 12) 8 ∙ 5 44 ≡8 (mod 12) 113 ≡5 (mod 12)
