9. Division Euclidienne et plus grand commun diviseur de deux entiers

a b b > 0q r
a=bq +r0≤r < a r
a b b
a b|a
a b
a b P GDC(a, b)
(a, b)
a, b ∈Z
x, y ∈Z
(a, b) = ax +by.
E={ax +by :x, y ∈Zax +by > 0} ⊂ N.
m=ax0+by0
m a b q 0≤r < m
a=mq +r.
r6= 0
r=a−mq =a−(ax0+by0)q=a(1 −x0q) + b(−y0q)∈E.
r < m m r = 0 m
a m b
m m0
a b p q a =pm0
b=qm0
m=ax0+by0=m0(px0+qy0).
N

m0m
x, y
a b (a, b) = 1
1< p ∈N1p
a, b ∈Zp
p|ab p|a p|b
Zn
nZ={nk :k∈Z}Z Z
a∼b⇐⇒ a−b∈nZ.
a b n
a≡b n.
Zn={[0],[1],· · · ,[n−1]}
[a] + [b]=[a+b]
n
Zna0≡a n
b0≡b n k, l ∈Za0=a+kn b0=b+ln
a0b0= (a+kn)(b+ln) = ab + (al +bk)n+kln2≡ab n.
[a] [b]
[a]·[b]=[ab].

Z6
A=Zn
[a]+[b] = [a+b] [a]·[b] = [ab]
A+
∀a, b, c ∈A, a(b+c) = ab +ac.
Zn
[a]∈Zn
[1][a]=[a][1] = [a].
Z6
[2][3] = [6] = [0].
Z6[5]
[5][5] = [1].
[a]∈Zn[a]
a n

a n
x, y ∈Z
ax +ny = 1.
ax = 1
n[1] = [ax]=[a][x].
Zn
Zn
n∈NU(n)
Zn
U(n) = {[a]∈Zn: (a, n) = 1}.
Z12
Z12
U(n)Zn
[1]
[1] U(n)
Z
ZnU(n)
[a],[b]∈U(n) [a0],[b0]∈Zn
[a][a0] = [1] [b][b0] = [1].
[a][b][b0][a0] = [a][1][a0]=[a][a0] = [1].
[a][b]∈U(n)
[a]∈U(n) [b]∈Zn[a][b] =
[1] [b]
[a]
[a][b]=[ab] = [ba]=[b][a]=1.

Z6
U(6) = {[1],[5]}.
6=2·3
1 5
p
06= [a]∈Zp
U(p) = {[1],[2],· · · ,[p−1]}