Ko nga nama tuatahi ko tera e rua noa o raatau whakawehe, i te mea ka wehe noa ratau i a raatau me te waahanga, ara, te nama 1. Engari kia tupato! Ka wehea ratau e nga tau pai me nga tau kino. He aha te tikanga o tenei? Tino ngawari. Ko te tau nui, hei tauira 2, ka taea anake te wehewehe ki te 2, -2, 1, me te -1.
Nga nama kua neke atu i te 2 nga kaiarahi e karangahia ana tau tito. Mena ka tangohia e tatou he tau hiato, hei tauira, 10, ka kite taatau ka taea te wehe i waenga i a ia me te kotahitanga, ara, i waenga i te 10 me te 1, engari kei waenga ano i te 2 me te 5. No reira, ko te 10 he tau hiato.
Ripanga Pihi
Ko nga tau katoa he mea nui, he whakahiatohe ranei?
E rua "motuhake" tau ehara i te mea nui, kaore ano hoki i te whakahiato: te 0 me te 1. Na te aha? Kia kite a:
- Ko te tau 1 ka taea te wehe takitahi (1/1 = 1) me te kotahitanga, ara ko te nama 1 (1/1 = 1). Heoi, mo te tau e kiia ana he pirimia, e rua pea nga momo wehewehe. Ko te nama 2 kotahi noa te kaiwhakarite, no reira ehara i te mea nui, kaore ano hoki i te hiato.
- Kaore e taea te wehe i te 0, na te mea kaore he whakatau.
Na, ki te tangohia e tatou te 0 me te 1 mai i te raarangi, mai i te maha o nga nama e toe ana, me pehea e mohio ai ko wai nga mea tuatahi ko ngahea kaore?
Me pehea te mohio mena he pirimia te tau
Ko te mea noa ko te whakaaro mo te mahi ma te maka atu, ara, te ngana ki te kimi i nga wehenga. Ma te tatauranga ka tere te tere, engari mena ka anga whakamuri, ma te pene me te pepa ranei, ka uaua haere nga mahi. Ka whakautuhia e maatau nga tikanga e rua kia mohio koe mena he nui te nama kaore ranei.
Te tatari o Eratosthenes
Ko te tatari o Eratosthenes he tikanga kia mohio ki nga nama tuatahi kei waenga i te 2, koinei te nama tuatahi, me tetahi tau.
Ko tenei tikanga ko te mahi teepu me te whakawhiti i nga tini o nga tau katoa. Tuatahi ka whakakorehia e maatau nga taarua o te 2, ka 3, ka peera tae noa ki te waa kua nui ake te tapawha i te nama whakamutunga o te teepu.
Ka rite ki nga mea katoa o te pangarau, ko te tatari Eratosthenes e tino maarama ana me tetahi tauira:
- Ka mahi maatau teepu me nga nama mai i te 2 ki te 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 |
- Ka tohaina e maatau nga taarua o te 2 mai i te raarangi, ara, ka whakawhiti atu i te 2 ki te 2: 4, 6, etc. Kia mataara! Ko te 2, ka taea noa te wehe i waenga i a ia me te nama 1, kaore maatau e whakawhiti atu, na te mea he tau matua.
| 2 | 3 | 5 | 7 | 9 | |||||
| 11 | 13 | 15 | 17 | 19 | |||||
| 21 | 23 | 25 | 27 | 29 |
- Ka tangohia e maatau te nama e whai ake nei, 3, ka tirohia he iti ake te tapawha i te nama nui rawa atu o te teepu. Hei 32 <30, ka haere tonu taatau ki te tatari ka whakawhiti i ona rahinga: 6, 9, 12 ... Ka rite ki nga mahi o mua, kaore e hipa i te nama 3, he mea nui hoki.
| 2 | 3 | 5 | 7 | ||||||
| 11 | 13 | 17 | 19 | ||||||
| 23 | 25 | 29 |
- Ka tuaruatia e maatau te hikoi o mua me te nama ka whai ake i te teepu: 4 kua whiti, na reira ka tangohia e 5. Hei 52 <30, ka whiti i taatau tini.
| 2 | 3 | 5 | 7 | ||||||
| 11 | 13 | 17 | 19 | ||||||
| 23 | 29 |
- Ka haere tonu maatau me te nama e whai ake nei kaore e whiti: 7. Hei 72 = 49, ara, ko te tapawha o te 7 he nui ake i te nama whakamutunga o te teepu, ka mutu nga tikanga, ko nga nama kaore e whakawhiti atu ko nga nama matua.
- Whakamutunga. Ko nga nama tuatahi kei waenga i te 2 me te 30 e: 2, 3, 5, 7, 11, 13, 17, 19, 23 me te 29.
Ko te tatari Eratosthenes he tere me te ngawari ki te mohio ki nga nama tuatahi, engari me peheahe aha mena ka nui rawa te nama e hiahia ana maatau ki te ako, hei tauira, 54657?
Ki taau e mohio ana, kaore he mahi ki te mahi teepu mai i te 2 ki te 54657, e tika? Me pehea e tatou? Tino ngawari: whakamahi paearu wehewehe.
Paearu wehewehe
Ko nga paearu wehewehe he ture kia kitea mena ka wehewehe tetahi tau i tetahi atu me te kore e wehewehe.
Na, ki te whakamahia e tatou enei ture me te maarama he wehewehe tetahi tau i tetahi atu nama ke atu i a ia me te waahanga, ka mohio taatau ehara i te mea nui.
- Te tohu o te wehewehe o te nama 2. Ko te tau ka wehea e te 2 mena he taurite, ara, mena ka mutu i te 0, 2, 4, 6 me te 8. Anei he tinihanga: penei i nga nama ka wehea e te 4, Ko te 6 me te 8 he mea wehe na te 2, kaore tatou e hiahia ki te mohio ki nga paearu wehewehe o era atu nama.
- Te tohu o te wehewehe o te nama 3. Ko te tau ka taea te wehe ki te 3 mena he maha te toru o te tapeke o ona mati. Kia tirohia te tauira:
267 -> 2 + 6 + 7 = 15
I te mea ko te 15 te maha o te 3, 267 ka wehea e te 3.
Hei taapiri, i te mea ko nga tau ka taea te wehe i te 9 ka wehea hoki e te 3, kaati ra ano mo taatau ki te mohio ki tenei paearu.
- Paearu wehewehe o te nama 5. Ko te tau e waatea ana e te 5 mena ka mutu he 0, he 5 ranei.
- Nga paearu mo te wehewehe o te nama 7. Kia kitea mena ka wehewehehia te tau e te 7, me tango e maatau te nama kaore te mati whakamutunga me te rua te mati whakamutunga. Mena ko te nama i riro ko te 0 he maha ranei o te 7, ko te tau tuatahi ka wehea e te 7. Ka maarama ake koe ki tenei me tetahi tauira, kia kaha!
378 -> 37 − (8 × 2) = 37 − 16 = 21
I te mea ko te 21 te maha o te 7, 378 ka wehea e te 7.
- Te tohu o te wehewehe o te nama 11. Mena ka tangohia e tatou te tapeke o nga nama taurangi me te tapeke o nga tau rereke, a, ko te nama i whiwhia he 0 te maha ranei o te 11, ko te tikanga ko te tau kua akohia ka wehea e te 11. Anei he tauira:
8591 -> (8 + 9) − (5 + 1) = 17 − 6 = 11
I te mea ko te 11 te maha o te 11, 8591 ka wehea e te 11.
Ana koira katoa! Kua tae ke atu ki a koe: ka mohio koe me pehea te tatau mena he mea tino nui taua tau 54657?
Rarangi o nga nama nui mai i te 1 ki te 10.000
Hei whakamutunga, mena kei te rapu koe i te raarangi nama tuatahi i waenga i te 1 ki te 10.000, penei i te 1 ki te 100 te 1 ki te 1.000 ranei, anei tetahi e oti ana, e whakahoutia ana.
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
