Arithmétique et congruences

m, n, p p 6= 0 m n p m ≡n[p]
m≡nmod p k m =kp +n
m p p 6= 0
q∈Zr06r < p m =qp +r q
m p r m p
m≡nmod p06n < p n m
p
a∈N
n≡mmod p a +n≡a+mmod p
n≡mmod p an ≡am mod p
x, y, x0, y0∈Zn, p ∈N
x≡ymod n x0≡y0mod n x +x0≡y+y0mod n
x·x0≡y·y0mod n
x≡ymod n xp≡ypmod n
P x ≡ymod n P (x)≡P(y) mod n
x≡ymod n x0≡y0mod n k k0x=y+kn x0=y0+k0n
x+x0= (y+y0)+(k+k0)n
p∈N1 1 ≡1 mod n1
xp
12! ≡ −1 mod 13
17561791
9 32111
nZ{m∈Z|m≡k
mod n}k n
2
n n
0,1, . . . , n −1nZ

E
k E ={m∈Z|m≡kmod n}` k n
E={m∈Z|m≡kmod `}`∈[[0, n −1]] n
n
E={m∈Z|m≡kmod k}F={m∈Z|m≡kmod `}k ` [[0, n −1]]
x∈E∩F x ≡kmod n x ≡`mod n
x n k ` k =` E =F E 6=F E ∩F=∅
m∈Zr m n m ∈ {x∈Z|x≡rmod n}
nZ
Z
Z
Z/nZn
k={m∈Z, m ≡kmod z}.Z/nZn{0, . . . , n −1}
n x +y x
y x y
x+yZ/nZX Y Z/nZ
X+Y x ∈X y ∈Y
x y
X−Y XY Z/nZ
Z Z/nZ
∀X, Y, Z ∈Z/nZ,(X+Y) + Z=X+ (Y+Z)
· ∀X, Y, Z ∈Z/nZ,(XY )Z=X(Y Z)
∀X, Y ∈Z/nZ, X +Y=Y+X
· ∀X, Y ∈Z/nZ, XY =Y X
0∀X∈Z/nZ, X + 0 = 0 + X=X
1· ∀X∈Z/nZ, X ·1 = 1 ·X=X
X−X={−x, x ∈X}n
X+ (−X) = 0
· ∀X∈Z/nZ,0·X=X·0 = 0
∀X, Y, Z ∈Z/nZ, X(Y+Z) = XY +XZ
Z1−1
3 5 3 ·2 = 6 ≡1 mod 5 3 ·2 = 1 3−1= 2
X Y 6= 0 XY = 0
Y X 0 = X−1XY = 1 ·Y=Y
Y6= 0 3 Z/6Z3·2 = 0
Z/6Z Z/7Z Z/9Z
n∈Zd∈N∗d n d n d0
n=d·d0
n1
1
d n n ≡0 mod d

d n d |n d n d -n
n∈N∗0
(a, b)∈(N∗)2a b a 6b
a x a y a αx +βy (α, β)∈Z2
d|n n |m d |m
p∈N∗p1
1 1
n1
n>2n= 2 2
n > 2 2 n−1
n
n d n 1n d ∈[[2, n −1]]
d n
n p1, . . . , pn
p=p1· · · pn+1. i ∈[[1, n]] p≡1 mod pip
pip1
p1, . . . , pn
n m
n m
m n
m n
1n m n
N
mn 1
m n d
d|m d |n d |pgcd(m, n)
m|d n |dppcm(m, n)|d
m, n ∈N∗m0m≡m0mod npgcd(m, n) = pgcd(m0, n)
m0m n
E m n E0
m0n
k m −m0=kn d ∈E d m kn m0d∈E0
E⊂E0E0⊂E E =E0
pgcd(m, n) = pgcd(m0, n)
m, n ∈N∗a b am +bn = pgcd(m, n)
nmin(m, n)∈N∗
min(m, n)=1 m= 1 n= 1 m= 1 pgcd(m, n)=1 a= 1
b= 0

k > 1 (m0, n0) min(m0, n0)< k a b
am0+bn = pgcd(m0, n) (m, n) min(m, n) = k
m=npgcd(m, n) = m a = 1 = 0
m > n r m n 16r < n 16r < m =k
(r, n)a0b0a0r+b0n=
pgcd(r, n) = pgcd(m, n)` m =`n +r
a0m+ (b0−a0`)n= pgcd(m, n)a=a0b=b0−a0`
m < n m n
a b c a c
c ab c b
a c u
v au +cv = 1 abu =b−cvb b =abu +cvb c ab abu c cvb
c b 
p p a p a
p p ab p a p b
n∈N∗
n
n= 1
n>2m
16m < n n p > 1
p01< p0< p p0n
p m =n
p16m < n
m n
p
n n =p1· · · pk=p0
1· · · p0
`
n p n p
pip p -pi
p n
p=p1
p=p0
1m=p2· · · pk=p0 2· · · p0
km m ∈[[1, n −1]]
k=`
i∈[[2, k]] pi=p0
in
n∈N∗
a b pgcd(a, b) = 1
ϕ n ∈N∗ϕ(n)m∈[[1, n −1]]
n
p ϕ(p)ϕ(pk)
m n m0m·m0≡1 mod n
m0∈Z/nZm·m0= 1 m0m
pZ/pZ
p(p−1)! ≡ −1 mod p