Number Theory: Modulus Math

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Bernice Mathews
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1 Page 1 of 5 How do you count? You might start counting from 1, or you might start from 0. Either way the numbers keep getting larger and larger; as long as we have the patience to keep counting, we could carry on forever No matter how long we keep going, the numbers will keep growing. That s because there are an infinite number of numbers... no chance to run out! Draw a number line. Why do we put arrows on both ends?... because the line keeps going forever. < > What would happen if we brought the ends of our number line together? Then the numbers would go in a circle Instead of continuing on forever like before, now our line comes back to where it started! Instead of having an infinite number of numbers, now we have a finite number of them! We call this special kind of arithmetic modulus math or modulus arithmetic. At first this may seem very strange, but in fact we use modulus arithmetic all the time without thinking of it. There are seven days in the week. When we get to the end of the week (Saturday), we don t go to Day Eight, but start counting again from Day One (Sunday). Likewise there are twelve months in the year. When we get to the last month of the year (December), we don t go looking for a thirteenth month but start again from the first (January)! 1. What other everyday uses of modulus arithmetic can you think of? 2. The above circle represents Modulo Eight (written mod8). Why do you think it is called Modulo Eight?

2 Page 2 of 5 3. Fill in the following table to find modulos Three, Five, Seven, Ten and Thirteen. You might find it helpful to draw a circle for each of them! Whole Numbers Modulo Modulo Modulo Modulo Modulo Notice that in modulus arithmetic, we normally start from 0, not 1. Why do you think this is so? Hint: it has to do with why modulus arithmetic is also known as the arithmetic of remainders! 5. Find x. You could, of course, use the mod function on your calculator to find the answer. (Did you always wonder what the mod key was for?) But try to find it without your calculator! a. x 110(mod 4) b. x 120(mod 5) c. x 130(mod 6) d. x 140(mod 7) e. x 150(mod 8) f. x 160(mod 9) g. x 170(mod 10) h. x 180(mod 11) i. x 190(mod 12) 6. Challenge question for the advanced students: Find x. Hint: x may have more than one solution! Another hint: try to factor! (If you re still lost, then look at one of the answers and try working backwards.) a. 7 95(mod x) b. 5 42(mod x) c. 6 51(mod x) d. 8 71(mod x) e. 4 73(mod x) f. 9 84(mod x) g. 3 17(mod x) h (mod x) i (mod x) 7. Find the first two (smallest) solutions for x. a. 5 = x(mod 6) b. 7 = x(mod 9) c. 13 = x(mod 17) 8. On what day of the week will Christmas fall in 2005? 2010? Hint: remember to take leap years into account. 9. In some cultures, it s very important to know on what day of the week you were born. For example, UN Secretary General Kofi Annan, who is from Ghana, is named Kofi because Kofi is the day name for Friday-born males in Ghana. I myself was born on March 5, When I lived in Ghana, my day name was Yaw. On what day of the week was I born? On what day of the week were you born? If you look on you will find your Ghanaian day name!

3 Page 3 of Find the sum of (1 + 2) (4 + 5) in Modulo What is the smallest value of x such that 7 + x = 3(mod 8)? 12. Copy and complete the table for multiplication in Modulo Seven on the set {1, 2, 3, 6} Using the above table, find x if x x 6 = 5(mod 7). 14. Find two possible values for x if x x = 4(mod 5). 15. Draw a table for addition in Modulo 9 on the set {1, 2, 3, 5, 7}. You can practice your modulus arithmetic by drawing tables for addition and multiplication in other modulos on the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

4 Page 4 of 5 ANSWERS TO QUESTIONS 1. Consider a clock: the hours start at 1 and go to 12 before they start over again at 1 (unless you live in parts of Europe, in which case the hours go all the way to 24 before starting over again at 1!). 2. Although the numbers only go up to 7, there are 8 of them because we count 0. Remember this for later when we talk about number bases and Base Eight! 3. Your table should look like this: Whole Numbers Modulo Modulo Modulo Modulo Modulo Find any number along the top row (the whole numbers ). Find any number along the first column (the modulos ). Find the intersection of the corresponding column and row. Then divide the whole number by the modulo number (for example, 5 for modulo 5) and take the remainder. It s the same number! What does this mean? If there is no remainder, then the value of that whole number in the corresponding modulo is 0. If there is a remainder, then the value of the whole number in the corresponding modulo is the same as the remainder a. x = 2 b. x = 0 c. x = 4 d. x = 0 e. x = 6 a. x = 8, 11, 88 b. x = 37 c. x = 9, 15, 45 d. x = 9, 21, 63 e. x = 23, 69 f. x = 7 g. x = 0 h. x = 4 i. x = 10 f. x = 15, 25, 75 g. x = 7, 14 h. x = 11, 121 i. x = 47

5 Page 5 of 5 7. a. x = 5, 11 b. x = 7, 16 c. x = 13, Sunday, Saturday 9. Thursday x = x = x = 2,

Grade 6/7/8 Math Circles April 1/2, Modular Arithmetic

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Modular Arithmetic Centre for Education in Mathematics and Computing Grade 6/7/8 Math Circles April 1/2, 2014 Modular Arithmetic Modular arithmetic deals