FICHE 3 Définition Grand Commun Diviseur) de a et de b et se note

a b
a b
a b P GCD
a b P GCD(a;b)
P GCD
a=bq +r P GCD(a;b) = P GCD(b;r)
d a b
d a −bq =r d
b r d b r
d a =bq +r a b
a b
b r P GCD(a;b) = P GCD(b;r)
b a P GCD(a;b) = b
b a b a b P GCD(a;b) = b
b b
P GCD(a;b) = b b a
a b
r1b r1r2
a=bq1+r10≤r1< b
b=r1q2+r20≤r2< r1
r1=r2q3+r30≤r3< r2
b r1r2r3
rn+1 = 0
rn−2=rn−1qn+rn0≤rn< rn−1
rn−1=rnqn+1.

P GCD(a;b) = P GCD(b;r1) = P GCD(r1;r2) = . . . =P GCD(rn−1;rn)
P GCD(rn−1;rn) = rn
P GCD a b rn
a b
rnP GCD
P GCD
a=bq1+r1138 807 = 52 089 ×2 + 34 629
b=r1q2+r252 089 = 34 629 ×1 + 17 460
r1=r2q3+r334 629 = 17 460 ×1 + 17 169
r2=r3q4+r417 460 = 17 169 ×1 + 291
r3=r4q517 169 = 291 ×59
P GCD r4= 291
P GCD(138 807 ; 52 089) = 291
a b
a b
P GCD
c c |a c |b
c|P GCD(a;b)
P GCD
P GCD(a;b) = P GCD(|a|;|b|)
a b k
P GCD(a;b) = P GCD(b;a)
P GCD(ka ;kb) = |k| × P GCD(a;b)
a b k P GCD µa
k;b k¶=P GCD(a;b) |k|

a b k (>0) k
P GCD
aa
kba k
P GCD
a b d =P GCD(a;b)a
d=a0b d=b0
P GCD(a
d;b d) = P GCD(a;b) d= 1
a b
a=a0d
b=b0d
P GCD(a0;b0) = 1
d=P GCD(a;b)
a b b 6= 0)
a
b
a
b
a0
b0
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P GCD
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a b
a= 300 b= 350
a= 168 b= 2 160
a= 308 b= 364
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a= 630 P GCD(a;b) = 105 600 < b < 1 100 b

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P GCD
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a b P GCD
a+b= 144
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n a = 2n2b=n(2n+ 1)
2n2n+ 1 P GCD a b
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a b a b ab = 1 734
P GCD a b