Lambobin Firayim sune wadanda suna da rabe 2 kawai, tun da su da rabe -raben ne kawai ke raba su, wato lamba 1. Amma a kula! Ana raba su ta lambobi masu kyau da marasa kyau. Menene ma'anar wannan? Mai sauqi. Babban lamba, misali 2, kawai za a iya raba shi da 2, -2, 1, da -1.
Ana kiran lambobi masu raba fiye da 2 hada lambobi. Idan muka ɗauki adadi mai lamba, misali, 10, za mu ga za mu iya raba shi da kansa da haɗin kai, wato tsakanin 10 zuwa 1, amma kuma tsakanin 2 zuwa 5. Saboda haka, 10 adadi ne.
Shin dukkan lambobi suna da mahimmanci ko a haɗe?
Akwai guda biyu lambobi "na musamman" wadanda ba su da mahimmanci ko mahadi: 0 da 1. Me ya sa? Bari mu gani:
- Za a iya raba lamba 1 da kanta (1/1 = 1) da haɗin kai, wato lamba 1 (1/1 = 1). Koyaya, don lambar da za a ɗauka azaman Firayim, dole ne ta sami rarrabuwa biyu daban -daban. Lambar 2 tana da rabe -raben guda ɗaya kawai, don haka ba babba ba ce ko kuma ta ƙunshi.
- Ba za a iya raba 0 da kanta ba, tunda sakamakon ba shi da iyaka.
Don haka idan muka cire 0 da 1 daga cikin jerin, daga cikin adadin sauran lambobin da suka rage, ta yaya zamu san wanne ne babba kuma wanene ba?
Yadda za a san idan lamba ta kasance babba
Abu mafi al'ada shi ne yin tunani game da yin ta ta hanyar jefar, wato ƙoƙarin nemo masu raba. Tare da kalkuleta yana da sauri sosai, amma idan dole ne mu yi shi a ƙasa ko tare da alkalami da takarda, abubuwa suna da rikitarwa. Muna koya muku hanyoyi biyu don sanin idan lamba ta fi kyau ko a'a.
Siera na Eratosthenes
Maganin Eratosthenes shine dabarun sanin manyan lambobi tsakanin 2, wanda shine lambar farko ta farko, da wani adadi.
Wannan hanya ta ƙunshi yin tebur da ƙetare yawan lambobi gaba ɗaya. Da farko za mu kawar da adadin 2, sannan 3, da sauransu har sai mun kai adadin da ke murabba'i ya fi lamba ta ƙarshe a kan tebur.
Kamar komai na lissafi, an fi fahimtar Eratosthenes sieve tare da misali:
- Muna yin tebur tare da lambobi daga 2 zuwa 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 |
- Muna ƙetare adadin 2 daga jerin, wato, muna ƙetare daga 2 zuwa 2: 4, 6, da dai sauransu. Yi hankali! Na 2, wanda kawai za a iya raba tsakaninsa da lambar 1, ba mu ƙetare ta ba, tunda ita ce lambar farko.
| 2 | 3 | 5 | 7 | 9 | |||||
| 11 | 13 | 15 | 17 | 19 | |||||
| 21 | 23 | 25 | 27 | 29 |
- Muna ɗaukar lamba ta gaba, 3, kuma duba cewa murabba'in bai kai adadin mafi girma a tebur ba. Kamar yadda 32 <30, muna ci gaba da sieve da ƙetare yawan sa: 6, 9, 12 ... Kamar yadda a matakin da ya gabata, ba mu ƙetare lamba 3, wanda kuma shine firayim.
| 2 | 3 | 5 | 7 | ||||||
| 11 | 13 | 17 | 19 | ||||||
| 23 | 25 | 29 |
- Muna maimaita mataki na baya tare da lamba ta gaba a cikin tebur: 4 an ƙetare, don haka muna ɗaukar 5. Kamar 52 <30, muna ƙetare yawan su.
| 2 | 3 | 5 | 7 | ||||||
| 11 | 13 | 17 | 19 | ||||||
| 23 | 29 |
- Muna ci gaba da lamba mai zuwa ba tare da tsallakawa ba: 7. Kamar 72 = 49, wato, murabba'in 7 ya fi lamba ta ƙarshe a cikin tebur, hanya ta ƙare, kuma lambobin da ba a haɗa su ba sune manyan lambobi.
- Kammalawa. Babban lambobin tsakanin 2 da 30 sune: 2, 3, 5, 7, 11, 13, 17, 19, 23 da 29.
Maganin Eratosthenes hanya ce mai sauri da sauƙi don sanin manyan lambobi, amma menenemenene idan lambar da muke so muyi karatu tayi yawa, misali, 54657?
Kamar yadda kuka fahimta, ba zai zama da amfani a yi tebur daga 2 zuwa 54657 ba, daidai ne? Me za mu iya yi a lokacin? Mai sauqi: yi amfani da ma'aunin rarrabuwa.
Ka'idojin rarrabuwa
Ka'idodin rarrabuwa sune ƙa'idoji don gano ko lambar ɗaya ta raba ta wani ba tare da yin rarrabuwa ba.
Don haka, idan muka yi amfani da waɗannan ƙa'idodin kuma muka lura cewa ana raba lamba ta wata lamba ban da ita da rukunin, za mu san cewa ba babba ba ne.
- Ma'anar rarrabuwa na lamba 2. Ana iya raba lamba da 2 idan ma, ma’ana, idan ta ƙare a 0, 2, 4, 6 ko 8. Kuma, a nan akwai dabara: kamar kowane lamba da za a iya raba ta 4, 6 ko 8 kuma ana raba su ta 2, ba za mu buƙaci sanin ƙa'idodin rarrabuwa na sauran ma lambobi ba.
- Ma'anar rarrabuwa na lamba 3. Ana iya raba lamba da 3 idan jimlar adadin sa ya ninka sau uku. Bari mu ga misali:
267 -> 2 + 6 + 7 = 15
Tunda 15 yana da lamba 3, 267 ana iya raba ta 3.
Bugu da kari, tunda kowane lamba da aka raba ta 9 shima ana raba shi da 3, zai ishe mu san wannan ma'aunin.
- Ma'anar rarrabuwa ta lamba 5. Ana iya raba lamba da 5 idan ta ƙare a 0 ko 5.
- Ma'anar rarrabuwa na lamba 7. Domin sanin ko lamba ta kasu kashi 7, dole ne mu cire lambar ba tare da lamba ta ƙarshe ba kuma sau biyu na ƙarshe. Idan lambar da aka samu ita ce 0 ko mahara 7, za a iya raba lambar farko ta 7. Za ku fahimci wannan da kyau tare da misali, bari mu kai gare ta!
378 -> 37 − (8 × 2) = 37 − 16 = 21
Tunda 21 yana da lamba 7, 378 ana iya raba ta 7.
- Ma'anar rarrabuwa na lamba 11. Idan muka cire jimlar adadin maɗaukaki da jimlar lambobi masu banƙyama, kuma lambar da aka samu ita ce 0 ko mahara na 11, wannan yana nufin cewa adadin binciken ya kasu kashi 11. A nan ne misali:
8591 -> (8 + 9) − (5 + 1) = 17 − 6 = 11
Tunda 11 yana da lamba 11, 8591 ana iya raba ta 11.
Kuma shi ke nan! Yanzu lokaci ya yi da ku: shin kun riga kun san yadda ake lissafi idan waccan babbar lambar, 54657, ita ce firayim?
Jerin manyan lambobi daga 1 zuwa 10.000
A ƙarshe, idan kuna neman jerin manyan lambobi tsakanin 1 zuwa 10.000, kamar 1 zuwa 100 ko 1 zuwa 1.000, ga cikakken da sabuntawa:
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
