Parimaths - Mathematical Olympiads Club Nombres p-adiques

p
p
p
qq
2 5
1 0,99999 . . .
9
a b a = 0, a1a2. . . b = 0, b1b2. . .
C(n)= 0, C(n)
1C(n) 2. . . C(n) 2nA(n)= 0, a1a2. . . anB(n)=
0, b1b2. . . bn
n≥1C(n)
n, C(n+1) n, . . .
C(∞)
n0, C(∞)
1C(∞) 2. . .
c=a×b
b
b≥2b
b

10
−1. . . 999
a b 10 10 a=. . . a2a1a0b=. . . b2b1b0
c=a×b10 10
c=. . . c2c1c0cncn−1. . . c0≡anan−1. . . a0×bnbn−1. . . b0(mod 10n+1)
p1
p10
10
10 10
10
p
p p
3
3 2
3−2