Bonjour,
1) (n-2) + (n-1) + n + (n+1) + (n+2) = 5n
et 5n ≡ 0 [10] ⇒ n pair
2) 555444333222111
= 111 x [10⁰ + 2x10³ + 3x10⁶ + 4x10⁹ + 5x10¹²]
111 ≡ 6 [7]
10 ≡ 3 [7] ⇒ 10³ ≡ 3³ ≡ -1 [7] ⇒ 10⁶ = (10³)² ≡ 1 [7], 10⁹ = (10³)³ ≡ (-1)³ = -1 [7] ...
Donc 555444333222111 ≡ 6 x (1 - 2 + 3 - 4 + 5) [7]
soit N ≡ 18 [7] ⇔ N ≡ 4 [7]
donc reste = 4
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Bonjour,
1) (n-2) + (n-1) + n + (n+1) + (n+2) = 5n
et 5n ≡ 0 [10] ⇒ n pair
2) 555444333222111
= 111 x [10⁰ + 2x10³ + 3x10⁶ + 4x10⁹ + 5x10¹²]
111 ≡ 6 [7]
10 ≡ 3 [7] ⇒ 10³ ≡ 3³ ≡ -1 [7] ⇒ 10⁶ = (10³)² ≡ 1 [7], 10⁹ = (10³)³ ≡ (-1)³ = -1 [7] ...
Donc 555444333222111 ≡ 6 x (1 - 2 + 3 - 4 + 5) [7]
soit N ≡ 18 [7] ⇔ N ≡ 4 [7]
donc reste = 4