ma103blect2 - Applied Algebra Lecture 2 Congruences Modular...

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1 Applied Algebra Lecture 2 - Congruences, Modular Arithmetic and Math. Induction March 30, 2010 Example. Clock Arithmetic Question. If it is 3 o’ clock now, what time is it after 163 hours? Method of Solution. Divide 163 by 12. Obtain the quotient 13 and the remainder 7. That is 163=13 12+7. Answer. It will be 10=7+3 o’clock. What if you want to know if it is A.M. or P.M.? Then you should divide by 24 rather than 12. We say that 3+163 is congruent to 10 modulo 12 and write 3+163 3+7 10 (mod 12). Gauss (see Gallian, p. 573) invented this notation. It gives us a new way to do arithmetic. This is modular arithmetic and is lies at the foundation of most of our applicatons. Defn. Fix a modulus m which is a positive integer. If a,b are integers, define a b(mod m) if and only if (iff) m divides (a-b) iff a and b have the same remainder upon division by m. Example. Let m=3. Draw a picture. We are taking the infinite line of integers and rolling it up into a 3-gon. -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 -5 -2 4 7 (mod 3) 2 -4 -1 5 8 (mod 3) 0 -3 0 3 6 9 (mod 3) Taking the modulus m=12, you get a clock. There are 3 congruence classes of integers mod 3. The set of these is 3 . We can identify 3 with {0,1,2} or with {-1,0,1}, or lots of other things.
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2 We can then use ordinary addition and multiplication of integers to define a sum and product on 3 . Example. 0+1 1(mod 3) 1+2 3 0(mod3) 2+2 4 1(mod 3) With this you get addition and multiplication tables for 3 which are also called Cayley tables , named for Cayley. See Gallian, p. 133. Cayley Table for Addition in 3 + 0 1 2 0 0 1 2 1 1 2 0 1+2 2 2 0 1 2+2 Cayley Table for Multiplication in 3 × 0 1 2 0 0 0 0 1 0 1 2 2 0 2 1 2 × 2 3
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