DIVISION EUCLIDIENNE

a b
(q;r)
a=bq +r0≤r < b
(a;b) (q;r)a
b q r a b
a
b
[bq ;b(q+ 1)[ bq ≤a<b(q+ 1) 0 ≤a−bq < b
r=a−bq
a= 114 b= 8 q= 14 r= 2
114 = 8 ×14 + 2 0 ≤r < 8
? b a r
?a
b=q+r b0≤r b<1qa bq=E³a b´
x n n ≤x < n + 1
n=E(x)
? r b −1b
b
b= 2 r= 0)
a= 2q r = 1)
a= 2q+ 1
b= 3
a= 3q a = 3q+ 1 a= 3q+ 2
b= 10 r a
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q r a b
a= 117 b= 28 a=−317 b= 21
a= 1 999 b= 4 a=−849 b= 13
a= 1 564 b= 156 a=−671 b= 6
a= 10 000 b= 11

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q100100 = 13q+35
100100 q
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n n2−1
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287 025 = 635 ×452 + 5
−287 025
−287 025
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a b
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a b
a b
a+b a −b
a2−b2
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n
n2+n+ 1 n+ 1
n2+n+ 1 n+ 2 n= 1 n6= 1)
2n−1 2n−1

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(x;y)
x+y= 6
2x+ 3y= 40 y
4x2+y2= 169
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(x;y)
xy =−6
x2−y2= 15
x2y+xy2= 12
4x2−y2= 20
(2x−3)(3y+ 1) = −2
(x−2)(y+ 3) = 14
x3+y3= 217
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(x;y)
xy −y+x= 12
xy = 15
2x2−xy −7 = 0
xy = 11 + x+ 2y
2xy = 14 + x+ 6y
x2−y2= 401
x2+ 57 = y2