[PDF] [PDF] Exercices bac -- 2011-2016 -- arithmétique E 1

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p q p q p-1≡1 (p) (un)ិ n u n=2n+3n+6n-1. n un n un (un) p

6×2p-2≡3 (p)6×3p-2≡2 (p)

6up-2≡0 (p)

p

11x-7y=5 xy

ិ (u;v)

11u-7v=1

(O;⃗ı,⃗ȷ) D

11x-7y-5=0 C M(x;y)

0⩽x⩽500⩽y⩽50

D C

11x2-7y2=5 xy

(x;y) x2≡2y2 xy x x2 y 2y2 x2 2y2 (x;y) xy xy (x;y) p a p ap-1≡ 1 (p) (un) ិ u

0=1 n,un+1=10un+21.

u1,u2u3 n,

3un=10n+1-7

n un u2 (un) n,un n, 3un≡4-(-1)n(11) n,un

1016≡1(17)

k,u16k+8 ab (u;v) ិau+bv=1 a,b,c a bc ab ac pq pq a a≡0 [p]a≡0 [q] a≡0 [pq]

S nិ

n≡9 [17] n≡3 [5] S (u;v) 17u+5v=1 ិ (u;v) n0=3×17u+9×5v n0 S n0 S

Sn S

n-n0≡0 [85] n S n=43+85kk

25x-108y=1xy

ិ (13 ; 3) a cg ិ 25g-108c=1 p a p ap-1 p ap-1≡1 [p] x x≡a[7]x≡a[19] x≡a[133] a a6≡1 [7] a108≡1 [7] (a25)g≡a[7] (a25)g≡a[7] a(a25)g≡a[19] (a25)g≡a[133] a 1⩽a⩽26 a r a25≡r[133]

0⩽r<133

rr1r13≡r1[133]0⩽r1<133ិ r1≡a[133]

128 59

a b c a b a⩾b ឹc ឹa a=13b=4 025
m

9m+526

p p ិ m ឹ p x 9x≡1 [26]

9m+5≡p[26]⇐⇒m≡3p-15 [26].

E 026

A A a→0,b→1,...z→25 a b c d e f g h i j k l m n o p q r s t u v w x y z⋆ a 0,b z 25 ⋆ 26
• xE g

4x+327

xE,g(x) E • g(x) s→18,g(18)=2121→v s v xE g(x)=x g x E y

E y≡4x+3 x≡7y+6

v f v e e e xy xy xy xy x 0ཝ y ཝ ឹxy ិ 8x+15y=1

8x+15y=146xy

AB A

AB A D AB D>0 B A A D B>A

D B-A

D A-B

ឹA

A=12B=14

AB

A=221B=331 ឹ

ិ (x;y) () 221x-331y=1. ិ (3 ; 2) (x;y) (un)(vn)ិ n u n=2+221n{ v 0=3 v n+1=vn+331 vn n (p;q) u p=vq, 0⩽p⩽5000⩽q⩽500 ABD ab r(a,b) ab c ab a b c̸=0 ឹb a=26b=9 abc ab ិ ab pq 025 x x′ិ x x′ p=9q=2 ឹ uv

9u+26v=1 ិ (u,v)

x′≡9x+2 [26] x≡3x′+20 [26] q=2p p p p=13q=2 (x,y,z) x

2+y2=z2.

32+42=9+16=25=52

(x,y,z) p (px,py,pz) (x,y,z) xyz n n=2α×kα k n=2α×k n

9=20×9,

120=23×15

192
xz x=2α×k z=2β×m 2x2z2

2x2 z2

(x,z) 2x2=z2 xyz x,y (x,y,z) xyz x4032=169×961

(x,z) z2-x2=4032 x<403 (x, 2015,z)

15x-26k=mxk

m ិ (u;v)

15u-26v=1

(x0;k0) (x;k) 15 (x-x0)-26(k-k0)=0 (x;k) x=26q+7m k=15q+4mq∈?. x xy 15x+7

26□□□ y

x y

15x+726

k 15x-26k=y-7 x=7y+326 n S(n) n ិ S(6)=12 S(7) n S(n)⩾1+n n S(n)=1+n np×qpq

S(n)=(1+p)(1+q)

nm

S(n×m)=S(n)×S(m)

npk p k n

S(n)=1-pk+1

1-p np13×q7 pq m mn st

0⩽s⩽130⩽t⩽7 m=ps×qt

S(n)=1-p14

1-p×1-q8

1-q 2n-1n a,bc (b;c)=1 baca bca 233-1

233-1)÷3

(233-1)÷4 (233-1)÷12 (233-1) (233-1) (233-1) (233-1)

2≡-1 [3] 233-1

S=1+23+(23)2+(23)3+···+(23)10

233-1

27-1 ិ (N,k)

Nk n k n (2n-1,k)̸=0k⩽p 2 n-1 ງk k>p 2 n-1 ឹ n=33 n=7 kឹ N p=11...1| {z p k=p-1∑ k=010k. p Np Np j10j≡13

Np≡p3

ឹ Np Np7 a {-3 ;-2 ;-1 ; 0 ; 1 ; 2 ; 3} 10m≡a7 m a p

10p≡17 p 6

p6ិ p Np=10p-19

Np 9Np

Np p

n 2 n≡... [10] n

2≡... [10]

m n=10m+1 n=10m-1 n2≡120 p Np p Np (a,b) (a,b) ab (O;⃗ı,⃗ȷ) ∆1 y=5 4 x-2 3 (x,y) 15x-12y 3 ∆1 ∆(E) :y=m n x-p q m,n,pq (m,n)=(p,q)=1 ឹ (E) ∆ ឹ m,n,pq ∆ (x0,y0)x0y0 ny0-mx0 q np qn qn (x0,y0) y0=m n x0-p q n=qrr uv qru-mv=1 (x0,y0) y 0=m n x0-p q ∆ y=38x-74

M,N,P,Q (M, N)

(P, Q)=1 X

M,N,P,Q

QN X 0 (M N X-P Q -M N X-P Q

X X+1

M N X-P Q ឹX,M N X-P Q ឹ-X,-M N X-P Q ិ M,N,P,Q (M, N) (P, Q)=1 a,bc a bc ab k bc=ka ab uv au+bv=1 c acu+bcv=c bcka acu+kav=c⇐⇒a(cu+kv)=c ac u1=2+3+6-1=10 u

2=4+9+36-1=48

u

3=8+27+216-1=250

u

4=16+81+1296-1=1392

u

5=32+243+7776-1=8050

u

6=64+729+46656-1=47448

2≡0 mod 2=⇒2n≡0 mod 2

3≡1 mod 2=⇒3n≡1 mod 2

6≡0 mod 2=⇒6n≡0 mod 2

un=2n+3n+6n-1≡0+1+0-1≡0 mod 2 u n u n n k∈?∗ n=2k

4≡0 mod 4, 4k≡0 mod 4

4 k×9k≡0 mod 4

9≡1 mod 4 9k≡1 mod 4

u

2k≡0+0+1-1=0 mod 4 u2k

u1 u2 u3 u5 p 2p-1≡1 modp

6×2p-2=3×2p-1⇐⇒3 modp

p, 3p-1≡1 modp

6×3p-2=2×3p-1≡2 modp

ិ 6up-2=6(2p-2+3p-2+6p-2-1)=

6×2p-2+6×3p-2+6p-1-6

6×2p-2≡3 modp 6×3p-2≡2 modp 6p-1≡1 modp

6×up-2≡3+2+1-6 modp6×up-2≡0 modp

6×up-2≡0 modp p

6×up-2 p pup-2

p (u;v) 11u-7v=1 11×2-7×3=1 (2 ; 3)

11×10-7×15=5 (10 ; 15)

(x;y) 11x-7y=11×10-7×15

11(x-10)=7(y-15).

11(x-10) x-10

k x-10=7k x-107k ិ y-15=11k (x;y) (x;y) (10+7k; 15+11k)k∈? ិ

D C 

x∈?;y∈?

11x-7y=5

0⩽x⩽50 ; 0⩽y⩽50⇔{

(x;y)

0⩽x⩽50 ; 0⩽y⩽50

k 0⩽10+7k⩽500⩽15+11k⩽50 -10 7 ⩽k⩽50 7 15 11 ⩽k⩽35 11 k-10123 C A(3 ; 4) B(10 ; 15) C(17 ; 26) D(24 ; 37) E(31 ; 48).

11≡1 (5)7≡2 (5)5≡0 (5) (x;y)

11x2-7y2=5x2-2y2≡0 (5)

x

2≡2y2(5)

x x2 y 2y2 x2 2y2 (x;y) x2≡2y2(5) x≡0 (5)y≡0 (5) xy xyab x=5ay=5b 11×25a2-7×25b2=5quotesdbs_dbs50.pdfusesText_50

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