Chapitre 2 - Arithmétique modulaire 1 Division euclidienne

a b b > 0
a b q r
a=b×q+r0≤r < b .
a b q r
23 4
23 = 4 ×5+3.
q=Ea
br=a−b×q
a b (q, r)
a=b×q+r0≤r < b .
b a a
b q = 0 r=a b a
b
q r

a b
•a b
•b a
q a =b×q
n n
•
•
•
x712x
•
•(35,56) =
a b
a b b
a b

(56,48) =
•
•
•
•
n≥2
n p
2≤p≤√n .

√133 ≈11.54
n≥2
n=pn1
1×pn2 2× ··· × pnk
k,
pini
56 =
•
•
•

a≡b[n].
•10 ≡31[3]
•16 ≡1[5]
•a a ≡0[2] a
n
a n ≥2
a≡0[n]
a a0b b0n
a≡a0[n]b≡b0[n].
a+b≡a0+b0[n]
a×b≡a0×b0[n]
ak≡(a0)k[n]
155−275
88−68