- ( 7- z ) ( dz 3 = at b) z : = xty Proof : distributivity ( A ) - - PDF document

* dla A dlb ⇒ d l atb - ASI : ( 2x ) ( dx - - a ) ( Ty ) ( dy = b ) - ( 7- z ) ( dz 3 € = at b) z : = xty Proof : distributivity ⇒ ( A ) - Same as sus ⇒ d la - B

x ly ¥ x Ey - 2/-4 - → F' Io # xsy NO 2710W % # xe YES 5/5

x - y 1 × 2 2 - y Find L W s . - . ' = x' ( x - y ) - y - w w=xty x - y l x' DO HH ' ' ' - y k 20 - Congruences DEI AI b mod m ← ifm/a-b Madd h E

75 mod 2 E x 2 1 75 - x 75=1 @ d 2) ' 100 I 0 ( mod 2) - ( Vn ) ( h E O need 2) n I I or acnode I → mod 3 2 I - l ( 3.) { O , I , 2 } • . mod 3 ) ) ( n I 0 C- n - 2 or I n or I I n IO

mod 6 { 0 , I , 2,3 , 4,5 } { o , It , I 2,3 } 3 =-3 (6) - Prove : DO p 25 if prime p , then mod 6 p III - - Gay . mod rn is a transitive relation

Conger mod m is a trans . . Assn . ( m )rdan a=b b IC m ) ( - D.C.a=e(m# translation m la - b =x ) Assn in lb . - c =y ' la - c -_zj - DC m - Z- Asg : mix h ly DC m II

Cony mod he is additive Assn cm ) a Ex biya DC ( m ) atb Exty - Assn mla - X = K m1b =L DC ml ( atb ) -4+53=14 - KtL=M a - b Ex - y ( m ) DC - Cx - g) =P Ml ca - Sl - K - L p -

Lemme b=yc# / is multi pl . Conger mod m Assn ( m ) a = x • § a b = xy ( m ) DC - Lemme If ) Cy e- x a t 't Thou art = x. t k ) - in 1 a - x Assn m I at - xt # la - x ) t DC what property of divisibility ? mla-xlca-xtttrafy.si } - ⇒ m I ca - x ) E

Hm , a ,x , t ⇒ at ( lemma : ( m ) a Ex ( ( l ) Assn m ) a = x b=yCm# I ab Exy ( m ) • DC ← § . - ( m ) a Ex • t :=b bae 'LL G) ⇒ ¥ . Cm ) by =yx=xy ( m ) (2) ⇒ - t :=x Lw ) use transitivity of congruence

④ Cm ) ⇒ ak=bkH a ⇒ - ⑤ - 1<=2 a -=b ( m ) ⇒ a2=b2 ( m ) - = - DC a=b Cm ) Assn a- ⇒ ' ( m ) a -=b ( m ) • T t - • anion ) a3cn Pf by induction on k