Nombre de zéros de n factoriel ; n petit⚓︎

On rappelle que, pour \(n\) un entier naturel, la factorielle de \(n\) se note \(n!\) et se définit comme le produit des entiers de \(1\) à \(n\).

\(0! = 1\), comme un produit vide.
\(1! = 1\)
\(2! = 1×2 = 2\)
\(3! = 1×2×3 = 6\)
\(11! = 1×2×3×4×5×6×7×8×9×10×11 = 39916800\)
\(42! = 1405006117752879898543142606244511569936384000000000\)

On constate que

\(3!\) se termine par aucun zéro.
\(11!\) se termine par 2 zéros.
\(42!\) se termine par 9 zéros.

Construire un tableau de longueur 1000, tel que nb_zeros_factorielle[n] contient le nombre de zéros dans l'écriture décimale de \(n!\), pour \(n\) entier inférieur à \(1000\).

Exemples

🐍 Console Python

>>> nb_zeros_factorielle[3]
0
>>> nb_zeros_factorielle[11]
2
>>> nb_zeros_factorielle[42]
9
>>> len(nb_zeros_factorielle) >= 1000
True

###

# testsbksl-nlbksl-nlassert nbpy-undzerospy-undfactorielle[3] == 0bksl-nlbksl-nlassert nbpy-undzerospy-undfactorielle[11] == 2bksl-nlbksl-nlassert nbpy-undzerospy-undfactorielle[42] == 9bksl-nlbksl-nlassert len(nbpy-undzerospy-undfactorielle) >= 1000bksl-nlbksl-nlbksl-nl# autres testsbksl-nlbksl-nlNBpy-undZEROS = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 31, 31, 31, 31, 31, 32, 32, 32, 32, 32, 33, 33, 33, 33, 33, 34, 34, 34, 34, 34, 35, 35, 35, 35, 35, 37, 37, 37, 37, 37, 38, 38, 38, 38, 38, 39, 39, 39, 39, 39, 40, 40, 40, 40, 40, 41, 41, 41, 41, 41, 43, 43, 43, 43, 43, 44, 44, 44, 44, 44, 45, 45, 45, 45, 45, 46, 46, 46, 46, 46, 47, 47, 47, 47, 47, 49, 49, 49, 49, 49, 50, 50, 50, 50, 50, 51, 51, 51, 51, 51, 52, 52, 52, 52, 52, 53, 53, 53, 53, 53, 55, 55, 55, 55, 55, 56, 56, 56, 56, 56, 57, 57, 57, 57, 57, 58, 58, 58, 58, 58, 59, 59, 59, 59, 59, 62, 62, 62, 62, 62, 63, 63, 63, 63, 63, 64, 64, 64, 64, 64, 65, 65, 65, 65, 65, 66, 66, 66, 66, 66, 68, 68, 68, 68, 68, 69, 69, 69, 69, 69, 70, 70, 70, 70, 70, 71, 71, 71, 71, 71, 72, 72, 72, 72, 72, 74, 74, 74, 74, 74, 75, 75, 75, 75, 75, 76, 76, 76, 76, 76, 77, 77, 77, 77, 77, 78, 78, 78, 78, 78, 80, 80, 80, 80, 80, 81, 81, 81, 81, 81, 82, 82, 82, 82, 82, 83, 83, 83, 83, 83, 84, 84, 84, 84, 84, 86, 86, 86, 86, 86, 87, 87, 87, 87, 87, 88, 88, 88, 88, 88, 89, 89, 89, 89, 89, 90, 90, 90, 90, 90, 93, 93, 93, 93, 93, 94, 94, 94, 94, 94, 95, 95, 95, 95, 95, 96, 96, 96, 96, 96, 97, 97, 97, 97, 97, 99, 99, 99, 99, 99, 100, 100, 100, 100, 100, 101, 101, 101, 101, 101, 102, 102, 102, 102, 102, 103, 103, 103, 103, 103, 105, 105, 105, 105, 105, 106, 106, 106, 106, 106, 107, 107, 107, 107, 107, 108, 108, 108, 108, 108, 109, 109, 109, 109, 109, 111, 111, 111, 111, 111, 112, 112, 112, 112, 112, 113, 113, 113, 113, 113, 114, 114, 114, 114, 114, 115, 115, 115, 115, 115, 117, 117, 117, 117, 117, 118, 118, 118, 118, 118, 119, 119, 119, 119, 119, 120, 120, 120, 120, 120, 121, 121, 121, 121, 121, 124, 124, 124, 124, 124, 125, 125, 125, 125, 125, 126, 126, 126, 126, 126, 127, 127, 127, 127, 127, 128, 128, 128, 128, 128, 130, 130, 130, 130, 130, 131, 131, 131, 131, 131, 132, 132, 132, 132, 132, 133, 133, 133, 133, 133, 134, 134, 134, 134, 134, 136, 136, 136, 136, 136, 137, 137, 137, 137, 137, 138, 138, 138, 138, 138, 139, 139, 139, 139, 139, 140, 140, 140, 140, 140, 142, 142, 142, 142, 142, 143, 143, 143, 143, 143, 144, 144, 144, 144, 144, 145, 145, 145, 145, 145, 146, 146, 146, 146, 146, 148, 148, 148, 148, 148, 149, 149, 149, 149, 149, 150, 150, 150, 150, 150, 151, 151, 151, 151, 151, 152, 152, 152, 152, 152, 156, 156, 156, 156, 156, 157, 157, 157, 157, 157, 158, 158, 158, 158, 158, 159, 159, 159, 159, 159, 160, 160, 160, 160, 160, 162, 162, 162, 162, 162, 163, 163, 163, 163, 163, 164, 164, 164, 164, 164, 165, 165, 165, 165, 165, 166, 166, 166, 166, 166, 168, 168, 168, 168, 168, 169, 169, 169, 169, 169, 170, 170, 170, 170, 170, 171, 171, 171, 171, 171, 172, 172, 172, 172, 172, 174, 174, 174, 174, 174, 175, 175, 175, 175, 175, 176, 176, 176, 176, 176, 177, 177, 177, 177, 177, 178, 178, 178, 178, 178, 180, 180, 180, 180, 180, 181, 181, 181, 181, 181, 182, 182, 182, 182, 182, 183, 183, 183, 183, 183, 184, 184, 184, 184, 184, 187, 187, 187, 187, 187, 188, 188, 188, 188, 188, 189, 189, 189, 189, 189, 190, 190, 190, 190, 190, 191, 191, 191, 191, 191, 193, 193, 193, 193, 193, 194, 194, 194, 194, 194, 195, 195, 195, 195, 195, 196, 196, 196, 196, 196, 197, 197, 197, 197, 197, 199, 199, 199, 199, 199, 200, 200, 200, 200, 200, 201, 201, 201, 201, 201, 202, 202, 202, 202, 202, 203, 203, 203, 203, 203, 205, 205, 205, 205, 205, 206, 206, 206, 206, 206, 207, 207, 207, 207, 207, 208, 208, 208, 208, 208, 209, 209, 209, 209, 209, 211, 211, 211, 211, 211, 212, 212, 212, 212, 212, 213, 213, 213, 213, 213, 214, 214, 214, 214, 214, 215, 215, 215, 215, 215, 218, 218, 218, 218, 218, 219, 219, 219, 219, 219, 220, 220, 220, 220, 220, 221, 221, 221, 221, 221, 222, 222, 222, 222, 222, 224, 224, 224, 224, 224, 225, 225, 225, 225, 225, 226, 226, 226, 226, 226, 227, 227, 227, 227, 227, 228, 228, 228, 228, 228, 230, 230, 230, 230, 230, 231, 231, 231, 231, 231, 232, 232, 232, 232, 232, 233, 233, 233, 233, 233, 234, 234, 234, 234, 234, 236, 236, 236, 236, 236, 237, 237, 237, 237, 237, 238, 238, 238, 238, 238, 239, 239, 239, 239, 239, 240, 240, 240, 240, 240, 242, 242, 242, 242, 242, 243, 243, 243, 243, 243, 244, 244, 244, 244, 244, 245, 245, 245, 245, 245, 246, 246, 246, 246, 246]bksl-nlbksl-nlfor n, attendu in enumerate(NBpy-undZEROS):bksl-nl assert nbpy-undzerospy-undfactorielle[n] == attendu, f"Erreur avec n = {n}"bksl-nlbksl-nl ∞/∞

nbpy-undzerospy-undfactorielle = [...]bksl-nl...bksl-nlbksl-nlbksl-nlbksl-nl# testsbksl-nlbksl-nlassert nbpy-undzerospy-undfactorielle[3] == 0bksl-nlbksl-nlassert nbpy-undzerospy-undfactorielle[11] == 2bksl-nlbksl-nlassert nbpy-undzerospy-undfactorielle[42] == 9bksl-nlbksl-nlassert len(nbpy-undzerospy-undfactorielle) >= 1000bksl-nlbksl-nlNone

Indice 1

Pour une version facile, on pourra utiliser une fonction qui renvoie le nombre de zéros d'un entier, on passera en argument la factorielle d'un entier.

Après avoir réussi. Tenter une version efficace avec l'indice 2.

Indice 2

Pour une version efficace, on cherchera à calculer l'augmentation du nombre de zéros, d'une factorielle à une autre, en fonction du nouveau facteur. Il s'agit du nombre de fois que l'on peut diviser par 5 ce nouveau facteur.