Feuille d`exercices 2 : corrigé

57 148 468 = 17 ×3 361 674 + 10 0 610 <17
57 148 468 17
3 361 674 10
84 279 736 = 17 ×4 957 630 + 26.
26 = 17 + 9
84 279 736 = 17 ×4 957 630 + 17 + 9 = 17 ×4 957 631 + 9.
069<17
84 279 736 17 4 957 631 9
A= 57 148 468 B= 84 279 736
A+B= 17 ×3 361 674 + 10 + 17 ×4 957 631 + 9
= 17 ×(3 361 674 + 4 957 631) + 19
= 17 ×8 319 305 + 17 + 2
= 17 ×8 319 306 + 2.
2<17
A+B17 8 319 306 2
2A= 2 ×(17 ×3 361 674 + 10)
= 17 ×(2 ×3 361 674) + 20
= 17 ×6 723 348 + 20
= 17 ×6 723 349 + 3.
3<17 2A
17 6 723 349 3
672−662−652+ 642= 4
n n −1n−2n−3
n2−(n−1)2−(n−2)2+ (n−3)2= 4.
26 >17 17 26

n n −1n−2n−3
n2−(n−1)2−(n−2)2+ (n−3)2
=n2−(n2−2n+ 1) −(n2−4n+ 4) + (n2−6n+ 9)
=n2−n2+ 2n−1−n2+ 4n−4 + n2−6n+ 9
= 4.
2n2n+ 2 n∈N
2n×(2n+ 2) = 2n×2(n+ 1) = 4n(n+ 1).
n(n+ 1)
n n(n+ 1)
n n + 1 n(n+ 1)
n(n+ 1) n(n+ 1) = 2 ×k
k4n(n+ 1) = 4 ×2×k= 8k
8
164 330 258 643
164 330 258 643
18
164 330 258 643
18 = 2 ×9
164 330 258 643
9
N N −1
N−1
10 = 2 ×5 2 5
N−1
N−1
60 N−1 = 60 N= 61 61
N−1
N−1 20 ×3 = 60

N−1
N−1
N−1 = 60 N= 61
=
=