Q Today is Thursday What day is it in 100 days 1020 days What day is it in Define I to be a nonzero real number let x c 112190 I such that X I 1 What is 2 2 What's OT 3 What have you known 2 as remsoothfied Discrete Mathematics and Applications by Kenneth Rosen We'll only work with 21 from Lec8toL
Primes and god 1 Recall Given a b C 21 a 10 we say a divides b written a lb if F C E Z s t b AC not both zero The largest d E I IDefI Let a b t Z s t d ta and d lb is called the greatestcommondivisory denoted gcd Ca b a and b of how do we find gcd a b Question Given such a and b Tmrw Fundamental Theorem of Arithmetic Every integer a product of primes can be uniquely written as b p Algorithm If a PEZ a p pkbk are prime Prak and b p then gcd pmainlak tpmincafbpzmincaz.ba a b factorization 120 123733 3 55 and 500 22 53 7214044 EI 2.3 year ne 230L 22.30 5 7 god 1 20,5 00 W 315L 5 Rene But prime factorization is very hard no efficient algorithm is known Find god 191,2871
K zbOqtrywhereaib.q r c 21 lLLm_ILet Then god Casad god Cbse exercise in discussion PI hm.TL TheDivisionAgorILhm het a d E Z and a I o Then there are unique q r c z with 0 Ergot a qd tr inreeal world in leacture such that remainder written a mood d r is the reminder Here d a or Tp m Algorithm 0 gang tab EI Find gcddEL.gl a 287cg x 3 9 1491g t 7 ILL x 6 2 KI I Is gcd 287,91 7 a b Bezout'stheorem If a b C It lThm then there exist coefficients SE t th s t E z such that ged Gib TT
Algorithm Run Euclidean algorithm backwards to get the coefficients This is called the extended Euclidean algorithm EI writegod 1287,91 as a linear combination of 287 and 91 KID HI 3 tie a EL x 6 Ia k x6 91 t 4 9ha Edoardo f 2 9t ILL 12 Ia I Is 91 goaliFind site 21 2 87 x 6 91 18 sit 7 2875 t 91 t 91 19 6 282 B T A T P Cd g Modular Arithmetic 2 IDef.TL Let a b t Z and MEE m 1 a b If we say or is congruenttobmodulomi denoted A b mod m 1007 2 mod 7 100 E 2 mod14 LEO 3 18 1417 mod 7 EI 2 100 12 1 41 3 mod 3 Il I Il T 1 mods 11 suggests it might be some Red The notation A EE b mod m sort of equality The following theorem tells us it is comparing reminders
IThm.TW Let a BEZ and MERT Then a b mod m iff e ra AIM boat rb Pf By divisionalgorithm F ga ab C 21,0 Era rb LM s t a gamtra b Abm rb A b ga ab m 1 ra rb i i I Assume a b mod m Then ml a b aem iira rnL OEE.IE mlcae m m I ra rb ra rb o okra m ra rb o ra Tb E Assume ra Rb Then a b ga 9 b m mta b b mod m a D 100 7 2,2 7 2 14 7 12 100 100 mod 7 Eg 2 In mod 3 11 4 3 I 11 1 11 3 1 1 3 1 100 2114
2 I additionand multiplication IThMI Let a b c Z and M C It b mod m and Ed If a modem mod m and AC Ibd Lm od m then at Fk c 21 s.t.my MI a b PI b mod m a b a MI C d D Fka EZ c d mod m s t make lost Mk t m122 Mlk 1124 late Cbt d m mod m b id at C mod m is similar left as an exercise bd Showing ac or 41h2 I either 41h2 O n mod 4 E.ge Prop.ILetnEZ.Tnh.enOn JoricmTd4j Eisa mode PI n Ees l 2,3 Notice that mod 4 Tb E 3 mod4 n I L'T n fi n 5 32 mod4 O or 1 mod 4 n
lprop.IT 4kt3formekEN 7m is not the sum of squares of two integers ba µ a't b for some a b t Z PI Suppose M od 4 of 0 or I Im o By previous prop mod 4 2 b O or 1 mi of mod 4 b O I 2 M 310141 However contradiction 4 k M 3 41M 3 a b dim.dm x y common arithmetic include them Given a c 112 mod m a Ebc X y X y My y yo X y D modem c additions and M ns preserve congruences Subtracting a C Z is the same as adding aE2j so subtractions preserve congruences at 21 is the same as multiplying ta Dividing But wait ta Ef 21 in general existence 2 2 Inverse unique Given at 21 MERT IDefI If L mod m KEI satisfies we say x is ax a inverseofamodulom.denoteda tmodu mRem.i is just a notation It is NOT the real number ta a We're only playing with 21 now remember
operation remainder a mod d a d Ek d Red a mod m c relationship AE b mod m A T M l a b An inverse of a modulo m mod m W a integer a EZ sit denotes a 1 L mod m a
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