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Circular Modular Addition and Multiplication Tables

Author:Devan, dborkovitz

Modular Arithmetic

Arithmetic dealing with the remainders after integers are divided by a fixed "modulus" m. Basically, it is a kind of integer arithmetic that reduces all numbers to ones that belongs to a fixed set [0 ... N-1]. We can say two integers, a and b, are congruent mod m (where m is a natural number) if both numbers divided by m produce the same remainder. In other words, m must evenly divide their difference, a - b.

Circular Patterns with Modular Arithmetic

Addition and Multiplication in Mod [6]

Addition and Multiplication in Mod [6]
Notice how the only numbers to appear in the tables below are 0, 1, 2, 3, 4, and 5. Any natural number, when divided by 6, will produce one of these 6 remainders. Notice from the table 5 + 5 = 4. This seems strange in the usual sense of addition we are used to, but notice that in mod 6 this is true. In fact, 5 + 5 = 10, and we know that 10 is congruent to 4 (mod 6). So, it is true 5 + 5 does actually equal 4! Similarly the table above tells us 5 * 5 = 1. Now this no longer comes as a surprise because we know 5 * 5 = 25, but 25 is actually congruent to 1 (mod 6). Therefore, 5 * 5 = 1! The tables above are accurate for addition and multiplication... in mod 6 of course!

Activity: Modular Arithmetic

1.) Practicing Addition: Do the following operations in Mod [8] a.) 6+3= ? b.) 12+4=? c.) 2+1=? 2.) Practicing Multiplication: Do the following operations in Mod [6] a.) 3 x 1=? b.) 3 x 6=? c.) 4 x 2=? 3.) Generation Patterns a.) explore the patterns formed from different combinations of Mods and operations b.) Fix the times at 2 , and increase the mod to the max and describe the pattern c.) Fix the times at 3, and increase the mod to the max and describe the pattern 4.) Extension: What patterns do you notice in 3a, and 3b? Is there a name for these patterns?

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