- ( 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 )

• 3€

(7-z) (dz

= at b)

Proof :

z : = xty

distributivity ⇒ ( A)

• Same assus ⇒ d la -B

x ly ¥

x Ey

2/-4

• → F'Io #

xsy

NO

2710W

% # xe

YES

5/5

x -y 1×2

• y

2

Find

W

s .-

L

(x -y)

• w

= x'

• y

'

w=xty

DO HH

x - y l x'

'

• y

''

k20

• Congruences

DEI AI b

mod m

←

ifm/a-b

h E

Madd

75

E x

mod 2

2 1 75 -x

75=1 @d 2)

• (Vn) ( h E O
• r

n I I

need2)

→

acnode I

mod 3

2 I - l

(3.)

{ O, I , 2}

.

• C-n

) ( nI 0

• r I

n

• 2

mod 3)

n IO

• r I I

mod 6

{0 , I , 2,3 , 4,5} { o , It

, I 2,3}

3 =-3 (6)

• DO

Prove :

if

p

prime

,

p 25

then

p III

mod6

• Gay

a

transitive relation

Conger

mod

m is a trans

Assn .

a=b

(m)rdan

b IC

(

m )

• D.C.a=e(m#

translation

Assn

m la - b =x ) in lb

• c =y
• DC

m

• Z-

Asg :

mix

h ly

DC

m II

Cony mod he

is additive

Assn

a Ex

cm)

biya

DC

atb Exty

( m)

• Assn

mla- X

= K

m1b =L

DC

ml (atb) -4+53=14

• KtL=M

DC

a - b Ex -y ( m)

Ml ca -Sl

• Cx -

g) =P

p

• K - L

Conger mod

m

is multipl .

• Assn

a = x

(m)

§

Lemme b=yc# /

DC

a b = xy

( m)

• Lemme If

a

e- x

Cy

)

t'tThou art = x.t k)

• Assn

in 1 a - x

DC

m I at -xt

#la -x)t

what property of divisibility ?

mla-xlca-xtttrafy.si}

• ⇒ m I ca -x)E

Hm , a,x ,t

lemma :

a Ex

(m)

⇒at(

Assn a = x

(

m )

(l)

b=yCm#I

• DC

ab Exy

( m)

§

. -

←

a Ex

( m)

• t :=b

G) ⇒ ¥

bae'LL

(2) ⇒

by =yx=xy (m)

• t :=x

Lw)

use transitivity

• f congruence

④

a ⇒

Cm) ⇒ ak=bkH

• ⑤ -

1<=2

a -=b ( m) ⇒ a2=b2 (m)

• =
• Assn

a=b Cm)

DC

• a -=b (m

)

a-⇒' ( m

)

T

t

• anion) a3cn

Pf

by induction on k