Projet Module INE22 Corrigé 1 Première partie : mise en jambes 2

a>0b > 0
(q, r)q>0 0 6r < b a =bq +r
a<b
(0, a)
q←0
r←a
r>b
q←q+ 1
r←r−b
(q, r)

r
(q, r)
q>0
06r < b
a=b∗q+r
a<b (0, a)
a=bq +r
r>0
q>0
r < b
q= 0 r=a
a=bq +r r >0q>0r>b
q0=q+ 1 r0=r−b q0r0
bq0+r0=b(q+ 1) + r−b=bq +r
r>b r0>0q>0q0
r
c=a d =b0< d 6c
r=c d 06r < d 2
1c=d×(c, d) + r n ∈Zn > 0d n = 0
r n = 0 d=b b n = 0 c=d×(c, d) + r
c=a3 4 a n = 0

a b 0< b 6a
n > 0a b
r←a b
r= 0
b
c←a
d←b
r > 0
c←d
d←r
r←c d
d
06r < d 0< d 6c∀n∈Z, n > 0d n = 0 r n =
0a n = 0 b n = 0 r > 0
c0=d d0=r r0=c0d0=d r c0d0
r0
r0=c0d02 0 6r0< d0
d0=r r > 0d0>0r < d c0=d d0< c0d06c0
n > 0d0n= 0 r0n= 0 d0=r
r n = 0 r0=d r 1d=
r×(d, r) + r03 4 d n = 0
n a n = 0 b n = 0
n
a b
a b = 0 b b > 0a b
6
n > 0d r
a b d d 6d r r = 0
5d a d b