Amanani aphambili ukusuka kwi-1 ukuya kwi-10.000

Amanani aphambili ngalawo bane-divider ezi-2 kuphela, kuba zahlulahlulwa ngokwazo kuphela yiyunithi, oko kukuthi, inani 1. Kodwa ke lumka! Ithetha ntoni le nto? Kulula kakhulu. Inombolo yokuqala, umzekelo 2, inokwahlulwa kuphela ngo-2, -2, 1, no -1.

Amanani angaphezu kwama-2 abahluli ababiziweyo amanani ahlanganisiweyo. Ukuba sithatha inani elidityanisiweyo, umzekelo u-10, siza kubona ukuba singakohlulahlula phakathi kwalo kunye nobunye, oko kukuthi, phakathi kwe-10 ne-1, kodwa kwakhona phakathi kwe-2 kunye ne-5.

Ngaba onke amanani aphambili okanye adityanisiwe?

Zimbini amanani "ezizodwa" ezingabalulekanga okanye ezimbaxa: i-0 kunye ne-1. Ngoba? Masibone:

• Inani 1 lungohlulwa lodwa (1/1 = 1) nangobunye, okt inani 1 (1/1 = 1). Nangona kunjalo, ukuze inani lithathelwe ingqalelo njengeyona iphambili, kufuneka libe nabathathi abahlukeneyo aba-2. Inombolo 1 inesahluli esinye kuphela, ke ayisiyonto iphambili okanye edibeneyo.
• I-0 ayinako ukwahlulwa yodwa, kuba iziphumo azingqinelani.

Ke ukuba sisusa u-0 no-1 kuluhlu, ngaphandle kwenani elikhulu lamanani ashiyekileyo, sazi njani ukuba zeziphi eziziiprayimari kwaye ezingezizo?

Wazi njani ukuba inani liphambili

Eyona nto iqhelekileyo kukucinga ngokwenza ngokulahla, Oko kukuthi, ukuzama ukufumana abahluleli. Ngekhaltyhuleyitha kuyakhawuleza, kodwa ukuba kufuneka siyenze sibheke ezantsi okanye ngepeni nephepha, izinto ziya zisiba nzima. Sikufundisa ngeendlela ezimbini zokwazi ukuba inani liyintloko okanye hayi.

Isihlunu se-Eratosthenes

Isihlunu se-Eratosthenes yi ubuchule bokwazi amanani aphambili phakathi kwe-2, elilelona nani lokuqala, kunye nenani elithile.

Le ndlela iqulethe ukwenza itafile kunye nokucanda okuphindaphindwayo kwamanani apheleleyo. Kuqala siza kuphelisa ukuphindaphindwa ngo-2, no-3, njalo-njalo de sifike kwinani eliphindwe kabini kunenani lokugqibela kwitafile.

Njengayo yonke into kwimathematics, i-Eratosthenes sieve iqondwa ngcono ngomzekelo:

1. Senza itafile ngamanani ukusuka ku-2 ukuya ku-30.

	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30

 

2. Siphinda-phinda isi-2 kuluhlu, oko kukuthi, siwela ukusuka ku-2 ukuya ku-2: 4, 6, njl. Lumka! I-2, enokwahlulwa phakathi kwayo kunye nenombolo 1, asiyicimi, kuba iyinombolo yokuqala.

	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30

 

3. Sithatha inani elilandelayo, 3, kwaye sijonga ukuba ophindwe kabini ungaphantsi kwelona nani likhulu kwitafile. Njengo-3² <30, siyaqhubeka ngesihlungulo kwaye sigqithisile kwiziphindaphindo zayo: 6, 9, 12 ... Njengakwisinyathelo esidlulileyo, asilicimi inani u-3, ekwaliphambili.

	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30

4. Siphinda inyathelo elidlulileyo ngenani elilandelayo kwitheyibhile: u-4 ugqityiwe, ngoko sithatha u-5 Njengo-5² <30, siwela ukuphindaphindeka kwabo.

	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30

5. Siyaqhubeka nenani elilandelayo ngaphandle kokucima: 7. Njengo 7² = 49, oko kukuthi, isikwere se-7 sikhulu kunenombolo yokugqibela etafileni, indlela iyaphela, kwaye amanani ngaphandle kokuwela ngaphandle ngamanani aphambili.
6. Isiphelo. Amanani aphambili phakathi kwe-2 kunye ne-30 ngala: 2, 3, 5, 7, 11, 13, 17, 19, 23 no-29.

Isihlunu se-Eratosthenes yindlela ekhawulezayo nelula yokwazi amanani aphambili, kodwa uthini ngawoungathini ukuba inani esifuna ukulifunda liphezulu kakhuluumzekelo, 54657?

Njengoko uqonda, ayinakuba yinto efanelekileyo ukwenza itafile ukusuka ku-2 kuye ku-54657, akunjalo? Yintoni esinokuyenza emva koko? Kulula kakhulu: sebenzisa iikhrayitheriya zokwahlulahlula.

Inkqubo yokwahlulahlula

Iikhrayitheriya zokwahlulahlula zi imigaqo yokufumanisa ukuba inani elinye lahlulwa lelinye ngaphandle kokwenza ukwahlula.

Ke, ukuba sisebenzisa le mithetho kwaye sijonga ukuba inani lahlulahlulwe lelinye inani ngaphandle kwalo kunye neyunithi, siyakwazi ukuba ayisiyonto iphambili.

• Ikhrayitheriya yokwahlulahlula inani u-2. Inombolo yahlula-hlulwe ngo-2 ukuba ngu-even, oko kukuthi, ukuba iphela ngo-0, 2, 4, 6 okanye 8. Kwaye, nali iqhinga: njengalo naliphi na inani elahluleke ngo-4, U-6 okanye u-8 ukwahlulwa-hlulwe ngo-2, asizukufuna ukwazi iikhrayitheriya zokwahlula amanye amanani.
• Ikhrayitheriya yokwahlulahlulwa kwamanani 3. Inombolo yahlulahlulwe ngo-3 ukuba isiphumo sokudityaniswa kwamanani asiphindaphindwe kathathu. Masibone umzekelo:

267 -> 2 + 6 + 7 = 15

Kuba u-15 uphinda-phinde ngo-3, 267 yahlulwe ngo-3.

Ukongeza, kuba onke amanani ahlule ngo-9 ukwahlula ngo-3, kuyakwanela kuthi ukuyazi le nqobo.

• Inani lokwahlula inombolo 5. Inombolo yahlulahlulwe ngesi-5 ukuba iphela ngo-0 okanye 5.
• Iikhrayitheriya zokwahlulwa-hlulwa kwenani 7. Ukufumanisa ukuba inani lahlula-hlulwe ngesi-7, kufuneka sikhuphe inani ngaphandle kwenani lokugqibela kwaye liphindwe kabini inani lokugqibela. Ukuba inani elifunyenweyo ngu-0 okanye uphinda-phinde ngo-7, inani lokuqala liyahlulwa ngo-7. Uyakukuqonda ngcono oku ngomzekelo, masifike kuwo!

378 -> 37 − (8 × 2) = 37 − 16 = 21

Kuba u-21 uphinda-phinde ngo-7, 378 yahlulwe ngo-7.

• Ikhrayitheriya yokwahlulahlulwa kwenani le-11. Ukuba sithabathe isambuku samanani alinganayo kunye nesambuku samanani angaqhelekanga, kwaye inani elifunyenweyo ngu-0 okanye uphinda-phinde ngo-11, oko kuthetha ukuba inani elifundwayo lahlulwe ngo-11. umzekelo:

8591 -> (8 + 9) − (5 + 1) = 17 − 6 = 11

Kuba u-11 uphinda-phinde ngo-11, 8591 yahlulwe ngo-11.

Kwaye konke oko! Ngoku lithuba lakho: ubusele uyazi ukuba ungabala njani ukuba inani eliphakamileyo, u-54657, yiprayimari?

Uluhlu lwamanani aphambili ukusuka kwi-1 ukuya kwi-10.000

Okokugqibela, ukuba ujonge uluhlu lwamanani aphambili phakathi kwe-1 ukuya kwi-10.000, ezinje nge-1 ukuya kwi-100 okanye i-1 ukuya kwi-1.000 XNUMX, nantsi epheleleyo nehlaziyiweyo:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919