. . . . "In der Mathematik ist eine Kaprekar-Zahl, benannt nach dem indischen Mathematiker D. R. Kaprekar, die Grundzahl a f\u00FCr eine Quadratzahl a\u00B2, deren Ziffernfolge (in einem Zahlensystem zur Basis b) in zwei Teile aufgeteilt werden kann, die miteinander addiert die Grundzahl a ergeben."@de . "\u30AB\u30D7\u30EC\u30AB\u30FC\u6570\uFF08\u30AB\u30D7\u30EC\u30AB\u30FC\u3059\u3046\u3001Kaprekar number\uFF09\u3068\u306F\u3001\u6B21\u306E\u3044\u305A\u308C\u304B\u3067\u5B9A\u7FA9\u3055\u308C\u308B\u81EA\u7136\u6570\u3067\u3042\u308B\u3002 1. \n* 2\u4E57\u3057\u3066\u4E0A\u4F4D\u306E\u534A\u5206\u3068\u4E0B\u4F4D\u306E\u534A\u5206\u3068\u306B\u5206\u3051\u3066\u548C\u3092\u53D6\u3063\u305F\u3068\u304D\u3001\u5143\u306E\u5024\u306B\u7B49\u3057\u304F\u306A\u308B\u81EA\u7136\u6570\u3002 2. \n* \u6841\u3092\u4E26\u3079\u66FF\u3048\u3066\u6700\u5927\u306B\u3057\u305F\u6570\u3068\u6700\u5C0F\u306B\u3057\u305F\u6570\u3068\u306E\u5DEE\u3092\u53D6\u3063\u305F\u3068\u304D\u3001\u5143\u306E\u5024\u306B\u7B49\u3057\u304F\u306A\u308B\u81EA\u7136\u6570\uFF08\u30AB\u30D7\u30EC\u30AB\u30FC\u5B9A\u6570\uFF09\u3002 \u540D\u79F0\u306F\u3001\u30A4\u30F3\u30C9\u306E\u6570\u5B66\u8005 D. R. \u30AB\u30D7\u30EC\u30AB\u30EB\uFF08\u82F1\u8A9E\u8868\u8A18: D. R. Kaprekar\uFF09\u306B\u3061\u306A\u3080\u3002\u30AB\u30D7\u30EC\u30AB\u6570\u3001\u30AB\u30D7\u30EA\u30AB\u6570\u3068\u3082\u3044\u3044\u3001\u539F\u8A9E\u3067\u3042\u308B\u30DE\u30E9\u30FC\u30C6\u30A3\u30FC\u8A9E\u306E\u767A\u97F3\u306B\u8FD1\u3065\u3051\u3066\u30AB\u30D7\u30EC\u30AB\u30EB\u6570\u3068\u3082\u3044\u3046\u3002"@ja . "N\u00FAmero de Kaprekar"@es . . "Kaprekartal"@sv . . "\u0427\u0438\u0441\u043B\u043E \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430 \u0434\u043B\u044F \u0434\u0430\u043D\u043D\u043E\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u044B \u0441\u0447\u0438\u0441\u043B\u0435\u043D\u0438\u044F \u2014 \u044D\u0442\u043E \u043D\u0435\u043E\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u0435 \u0446\u0435\u043B\u043E\u0435 \u0447\u0438\u0441\u043B\u043E, \u043A\u0432\u0430\u0434\u0440\u0430\u0442 \u043A\u043E\u0442\u043E\u0440\u043E\u0433\u043E \u0432 \u044D\u0442\u043E\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u0435 \u043C\u043E\u0436\u043D\u043E \u0440\u0430\u0437\u0431\u0438\u0442\u044C \u043D\u0430 \u0434\u0432\u0435 \u0447\u0430\u0441\u0442\u0438, \u0441\u0443\u043C\u043C\u0430 \u043A\u043E\u0442\u043E\u0440\u044B\u0445 \u0434\u0430\u0451\u0442 \u0438\u0441\u0445\u043E\u0434\u043D\u043E\u0435 \u0447\u0438\u0441\u043B\u043E. \u041D\u0430\u043F\u0440\u0438\u043C\u0435\u0440, 45 \u044F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u0447\u0438\u0441\u043B\u043E\u043C \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430, \u043F\u043E\u0441\u043A\u043E\u043B\u044C\u043A\u0443 452 = 2025 \u0438 20 + 25 = 45. \u0427\u0438\u0441\u043B\u0430 \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430 \u043D\u0430\u0437\u0432\u0430\u043D\u044B \u043F\u043E \u0438\u043C\u0435\u043D\u0438 \u0414. \u0420. \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430."@ru . "Kaprekar number"@en . . . "In matematica, un numero di Kaprekar in una data base \u00E8 un numero intero non-negativo, il cui quadrato (nella stessa base) pu\u00F2 essere suddiviso in due parti che, sommate tra loro, forniscono nuovamente il numero di partenza. La riformulazione dei concetti esposti in termini pi\u00F9 rigorosi pu\u00F2 essere cos\u00EC espressa: Si consideri un numero X che sia intero e non negativo. X \u00E8 un numero di Kaprekar in base b se esistono dei numeri interi non negativi n, A e B che soddisfino le tre condizioni seguenti: 0 < B < bnX\u00B2 = Abn + BX = A + B I primi numeri di Kaprekar in base 10 sono:"@it . . "\u5361\u5E03\u5217\u514B\u6578\uFF08Kaprekar number\uFF09\u662F\u5177\u6709\u4EE5\u4E0B\u6027\u8CEA\u7684\u6578\uFF1A \u5C0D\u65BC\u67D0\u500B\u6B63\u6574\u6578\u5728n\u9032\u4F4D\u4E0B\u5B58\u5728\u6B63\u6574\u6578 A, B \u53CA m\uFF0C\u4E14 \n* \n* \n* \u7C21\u55AE\u7684\u8AAA\uFF0C\u82E5\u6B63\u6574\u6578X\u5728n\u9032\u4F4D\u4E0B\u7684\u5E73\u65B9\u53EF\u4EE5\u5206\u5272\u70BA\u4E8C\u500B\u6578\u5B57\uFF0C\u800C\u9019\u4E8C\u500B\u6578\u5B57\u76F8\u52A0\u5F8C\u6070\u7B49\u65BCX\uFF0C\u90A3\u9EBCX\u5C31\u662Fn\u9032\u4F4D\u4E0B\u7684\u5361\u5E03\u5217\u514B\u6578\u3002 \u4F8B\u5982 297 \u5728\u5341\u9032\u4F4D\u4E0B\u662F\u5361\u5E03\u5217\u514B\u6578\uFF0C\u56E0\u70BA\uFF0C\u53EF\u4EE5\u5206\u5272\u6210 88 \u53CA 209\uFF0C\u4E14 88+209=297\u3002\u4E0D\u904E 100 \u5728\u5341\u9032\u4F4D\u4E0B\u4E0D\u662F\u5361\u5E03\u5217\u514B\u6578\uFF0C\u96D6\u7136\uFF0C\u53EF\u4EE5\u5206\u5272\u6210 100 \u53CA 00\uFF0C\u4F46 00 \u4E0D\u662F\u6B63\u6574\u6578\u3002 \u5728\u5341\u9032\u4F4D\u4E0B\uFF0C\u5E7E\u500B\u8F03\u5C0F\u7684\u5361\u5E03\u5217\u514B\u6578\u5982\u4E0B\uFF08OEIS\u6578\u5217\uFF09: 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170 \u5728\u4E8C\u9032\u4F4D\u4E0B\uFF0C\u6240\u6709\u7684\u5B8C\u5168\u6578\u90FD\u662F\u5361\u5E03\u5217\u514B\u6578\u3002 \u6709\u65F6\u5019\uFF0C\u4EBA\u4EEC\u4F1A\u628A6174\u8FD9\u4E2A\u6570\u79F0\u4F5C\u5361\u5E03\u5217\u514B\u6578\uFF0C\u800C\u5176\u5B9E\u8FD9\u662F\u5361\u5E03\u5217\u514B\u5E38\u6570\u3002"@zh . . "17697"^^ . . . . . "\uCE74\uD504\uB9AC\uCE74 \uC218"@ko . "\u0427\u0438\u0441\u043B\u043E \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430 \u0434\u043B\u044F \u0434\u0430\u043D\u043E\u0457 \u0441\u0438\u0441\u0442\u0435\u043C\u0438 \u0447\u0438\u0441\u043B\u0435\u043D\u043D\u044F - \u0446\u0435 \u043D\u0435\u0432\u0456\u0434'\u0454\u043C\u043D\u0435 \u0446\u0456\u043B\u0435 \u0447\u0438\u0441\u043B\u043E, \u043A\u0432\u0430\u0434\u0440\u0430\u0442 \u044F\u043A\u043E\u0433\u043E \u0432 \u0446\u0456\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u0456 \u0447\u0438\u0441\u043B\u0435\u043D\u043D\u044F \u043C\u043E\u0436\u043D\u0430 \u0440\u043E\u0437\u0431\u0438\u0442\u0438 \u043D\u0430 \u0434\u0432\u0456 \u0447\u0430\u0441\u0442\u0438\u043D\u0438, \u0441\u0443\u043C\u0430 \u044F\u043A\u0438\u0445 \u0434\u0430\u0454 \u043F\u043E\u0447\u0430\u0442\u043A\u043E\u0432\u0435 \u0447\u0438\u0441\u043B\u043E. \u041D\u0430\u043F\u0440\u0438\u043A\u043B\u0430\u0434, 45 \u0454 \u0447\u0438\u0441\u043B\u043E\u043C \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430, \u043E\u0441\u043A\u0456\u043B\u044C\u043A\u0438 452 = 2025 \u0456 20 + 25 = 45. \u0427\u0438\u0441\u043B\u0430 \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430 \u043D\u0430\u0437\u0432\u0430\u043D\u0456 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C ."@uk . "Em matem\u00E1tica, um n\u00FAmero de Kaprekar numa determinada base \u00E9 um inteiro n\u00E3o-negativo, tal que a representa\u00E7\u00E3o do seu quadrado nessa base pode ser dividida em duas partes que somadas permitem obter o n\u00FAmero original (ao dividir o n\u00FAmero cujas partes voc\u00EA vai adicionar, deixe a parte mais longa \u00E0 direita). Por exemplo, 297 \u00E9 um n\u00FAmero de Kaprekar para a base 10, porque 297\u00B2 = 88209, que pode ser dividido em 88 e 209, e 88 + 209 = 297. A segunda parte pode come\u00E7ar pelo algarismo 0, mas tem de ser um n\u00FAmero positivo. Por exemplo, 999 \u00E9 um n\u00FAmero de Kaprekar para a base 10, porque 999\u00B2 = 998001, que se separa em 998 e 001, e 998 + 001 = 999. Mas 100 n\u00E3o \u00E9 um n\u00FAmero de Kaprekar; embora 100\u00B2 = 10000 e 100 + 00 = 100, a segunda parte n\u00E3o \u00E9 positiva."@pt . "Em matem\u00E1tica, um n\u00FAmero de Kaprekar numa determinada base \u00E9 um inteiro n\u00E3o-negativo, tal que a representa\u00E7\u00E3o do seu quadrado nessa base pode ser dividida em duas partes que somadas permitem obter o n\u00FAmero original (ao dividir o n\u00FAmero cujas partes voc\u00EA vai adicionar, deixe a parte mais longa \u00E0 direita). Por exemplo, 297 \u00E9 um n\u00FAmero de Kaprekar para a base 10, porque 297\u00B2 = 88209, que pode ser dividido em 88 e 209, e 88 + 209 = 297. A segunda parte pode come\u00E7ar pelo algarismo 0, mas tem de ser um n\u00FAmero positivo. Por exemplo, 999 \u00E9 um n\u00FAmero de Kaprekar para a base 10, porque 999\u00B2 = 998001, que se separa em 998 e 001, e 998 + 001 = 999. Mas 100 n\u00E3o \u00E9 um n\u00FAmero de Kaprekar; embora 100\u00B2 = 10000 e 100 + 00 = 100, a segunda parte n\u00E3o \u00E9 positiva. Matematicamente exposto, seja X um inteiro n\u00E3o-negativo. X \u00E9 um n\u00FAmero de Kaprekar para a base b quando existem inteiros n\u00E3o-negativos n, A e B que satisfa\u00E7am as tr\u00EAs seguintes condi\u00E7\u00F5es: 0 < B < bnX\u00B2 = Abn + BX = A + B Os primeiros n\u00FAmeros de Kaprekar na base 10 s\u00E3o (sequ\u00EAncia na OEIS): 1, 9, 45, 55, 99, 297, 703, 999 , 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 643357, 648648, 670033, 681318, 791505, 812890, 818181, 851851, 857143, 961038, 994708, 999999. No sistema de numera\u00E7\u00E3o bin\u00E1rio todos os n\u00FAmeros perfeitos pares s\u00E3o n\u00FAmeros de Kaprekar."@pt . . . . . "\u064A\u0633\u0645\u0649 \u0639\u062F\u062F \u0645\u0627 \u0639\u062F\u062F \u0643\u0627\u0628\u0631\u0643\u0627\u0631 (\u0628\u0627\u0644\u0625\u0646\u063A\u0648\u0634\u064A\u0629: kaprekar number). \u0625\u0630\u0627 \u062D\u0642\u0642 n \u0627\u0644\u0634\u0631\u0637 \u0627\u0644\u062A\u0627\u0644\u064A: n\u00B2 \u064A\u0643\u062A\u0628 \u0641\u064A \u0634\u0643\u0644 \u0643\u062A\u0627\u0628\u0629 \u0630\u0627 \u0642\u0627\u0639\u062F\u0629 n (\u0639\u0627\u062F\u0629 \u0639\u0634\u0631\u0629) (\u0645\u062B\u0644\u0627:abcd). \u062A\u0642\u0633\u0645 \u0627\u0644\u0643\u062A\u0627\u0628\u0629 \u0625\u0644\u0649 \u062C\u0632\u0623\u064A\u0646 (ab , cd), \u0645\u062C\u0645\u0648\u0639 \u0627\u0644\u062C\u0632\u0623\u064A\u0646 (ab + cd) \u064A\u062C\u0628 \u0623\u0646 \u064A\u0633\u0627\u0648\u064A n \n* \u0645\u062B\u0644\u0627 9 \u0647\u0648 \u0639\u062F\u062F \u0643\u0627\u0628\u0631\u064A\u0643\u0627\u0631 \u0644\u0623\u0646 9\u00B2=81 \u0648 8 + 1 = 9 \n* \u0625\u0630\u0627 \u0643\u0627\u0646 \u0639\u062F\u062F \u0623\u0631\u0642\u0627\u0645 \u0645\u0631\u0628\u0639 \u0627\u0644\u0639\u062F\u062F \u0641\u0631\u062F\u064A\u0627. \u064A\u0643\u0648\u0646 \u0639\u062F\u062F \u0623\u0631\u0642\u0627\u0645 \u0627\u0644\u0646\u0635\u0641 \u0627\u0644\u064A\u0645\u064A\u0646\u064A \u0623\u0643\u0628\u0631 \u0628\u0648\u0627\u062D\u062F \u0645\u0646 \u0627\u0644\u0646\u0635\u0641 \u0627\u0644\u0622\u062E\u0631 \u0645\u062B\u0627\u0644: 9 \u0647\u0648 \u0639\u062F\u062F \u0643\u0627\u0628\u0631\u064A\u0643\u0627\u0631 \u0644\u0623\u0646 297\u00B2=88209 \u0648 88 + 209 = 297 \n* \u0627\u0644\u0623\u0639\u062F\u0627\u062F \u0627\u0644\u0643\u0627\u0628\u0631\u064A\u0643\u0627\u0631 \u0627\u0644\u0623\u0648\u0644\u0649 \u0647\u064A 1,0, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170... \n* \u0628\u0645\u0627 \u0623\u0646 \u062E\u0627\u0635\u064A\u0629 \u0627\u0644\u0639\u062F\u062F \u0643\u0627\u0628\u0631\u064A\u0643\u0627\u0631 \u0645\u062A\u0639\u0644\u0642\u0629 \u0628\u0623\u0631\u0642\u0627\u0645\u0647 \u0641\u0625\u0646 \u0627\u0644\u0623\u0639\u062F\u0627\u062F \u0627\u0644\u0643\u0627\u0628\u0631\u064A\u0643\u0627\u0631 \u062A\u062E\u062A\u0644\u0641 \u0628\u0627\u062E\u062A\u0644\u0627\u0641 \u0646\u0638\u0627\u0645 \u0627\u0644\u0639\u062F."@ar . . . . . . . . "In matematica, un numero di Kaprekar in una data base \u00E8 un numero intero non-negativo, il cui quadrato (nella stessa base) pu\u00F2 essere suddiviso in due parti che, sommate tra loro, forniscono nuovamente il numero di partenza. Per esempio, 297 \u00E8 un numero di Kaprekar nel sistema numerico decimale, perch\u00E9 2972 = 88209, che si pu\u00F2 suddividere in 88 e 209, e 88 + 209 = 297. La seconda parte pu\u00F2 iniziare con uno zero, ma deve essere un numero positivo. Per esempio, 999 \u00E8 un numero di Kaprekar in base 10, poich\u00E9 9992 = 998001, che si pu\u00F2 dividere in 998 e 001, e 998 + 001 = 999 mentre il numero 100 non lo \u00E8 poich\u00E9 anche se 1002 = 10000 e 100 + 00 = 100, il secondo addendo non \u00E8 un numero positivo. La riformulazione dei concetti esposti in termini pi\u00F9 rigorosi pu\u00F2 essere cos\u00EC espressa: Si consideri un numero X che sia intero e non negativo. X \u00E8 un numero di Kaprekar in base b se esistono dei numeri interi non negativi n, A e B che soddisfino le tre condizioni seguenti: 0 < B < bnX\u00B2 = Abn + BX = A + B I primi numeri di Kaprekar in base 10 sono: 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170 Nella numerazione binaria, tutti i numeri perfetti pari sono numeri di Kaprekar. Per ogni base esistono infiniti numeri di Kaprekar; in particolare, per una data base b tutti i numeri di forma bn - 1 sono numeri di Kaprekar. I numeri di Kaprekar prendono il nome da D. R. Kaprekar."@it . "In der Mathematik ist eine Kaprekar-Zahl, benannt nach dem indischen Mathematiker D. R. Kaprekar, die Grundzahl a f\u00FCr eine Quadratzahl a\u00B2, deren Ziffernfolge (in einem Zahlensystem zur Basis b) in zwei Teile aufgeteilt werden kann, die miteinander addiert die Grundzahl a ergeben."@de . . "En matem\u00E0tiques, els nombres de Kaprekar s\u00F3n nombres naturals que satisfan la condici\u00F3 de que el seu quadrat es pot tallar en dos trossos que, sumats, donen el nombre original. Formalment, doncs, s\u00F3n nombres naturals, , que satisfan les equacions: Aquest nombres foren introduits el 1980 pel matem\u00E0tic indi D. R. Kaprekar. Per exemple, \u00E9s un nombre de Kaprekar perqu\u00E8: i : tamb\u00E9 ho \u00E9s per i . Altres casos son m\u00E9s dif\u00EDcils de trobar: i . A continuaci\u00F3 es mostren altres exemples: La s\u00E8rie OEIS 6886 mostra tots els nombres de Kaprekar en base 10 elevats al quadrat. , en la qual: i"@ca . . . . . . . . . "Let\n\n\n\nThen,\n\n\n\nThe two numbers and are\n: \n: \nand their sum is\n\n\n\nThus, is a Kaprekar number."@en . . . "\u30AB\u30D7\u30EC\u30AB\u30FC\u6570"@ja . . . . . . . . . . . "1122574822"^^ . . "\u30AB\u30D7\u30EC\u30AB\u30FC\u6570\uFF08\u30AB\u30D7\u30EC\u30AB\u30FC\u3059\u3046\u3001Kaprekar number\uFF09\u3068\u306F\u3001\u6B21\u306E\u3044\u305A\u308C\u304B\u3067\u5B9A\u7FA9\u3055\u308C\u308B\u81EA\u7136\u6570\u3067\u3042\u308B\u3002 1. \n* 2\u4E57\u3057\u3066\u4E0A\u4F4D\u306E\u534A\u5206\u3068\u4E0B\u4F4D\u306E\u534A\u5206\u3068\u306B\u5206\u3051\u3066\u548C\u3092\u53D6\u3063\u305F\u3068\u304D\u3001\u5143\u306E\u5024\u306B\u7B49\u3057\u304F\u306A\u308B\u81EA\u7136\u6570\u3002 2. \n* \u6841\u3092\u4E26\u3079\u66FF\u3048\u3066\u6700\u5927\u306B\u3057\u305F\u6570\u3068\u6700\u5C0F\u306B\u3057\u305F\u6570\u3068\u306E\u5DEE\u3092\u53D6\u3063\u305F\u3068\u304D\u3001\u5143\u306E\u5024\u306B\u7B49\u3057\u304F\u306A\u308B\u81EA\u7136\u6570\uFF08\u30AB\u30D7\u30EC\u30AB\u30FC\u5B9A\u6570\uFF09\u3002 \u540D\u79F0\u306F\u3001\u30A4\u30F3\u30C9\u306E\u6570\u5B66\u8005 D. R. \u30AB\u30D7\u30EC\u30AB\u30EB\uFF08\u82F1\u8A9E\u8868\u8A18: D. R. Kaprekar\uFF09\u306B\u3061\u306A\u3080\u3002\u30AB\u30D7\u30EC\u30AB\u6570\u3001\u30AB\u30D7\u30EA\u30AB\u6570\u3068\u3082\u3044\u3044\u3001\u539F\u8A9E\u3067\u3042\u308B\u30DE\u30E9\u30FC\u30C6\u30A3\u30FC\u8A9E\u306E\u767A\u97F3\u306B\u8FD1\u3065\u3051\u3066\u30AB\u30D7\u30EC\u30AB\u30EB\u6570\u3068\u3082\u3044\u3046\u3002"@ja . . . . . . . . "\u064A\u0633\u0645\u0649 \u0639\u062F\u062F \u0645\u0627 \u0639\u062F\u062F \u0643\u0627\u0628\u0631\u0643\u0627\u0631 (\u0628\u0627\u0644\u0625\u0646\u063A\u0648\u0634\u064A\u0629: kaprekar number). \u0625\u0630\u0627 \u062D\u0642\u0642 n \u0627\u0644\u0634\u0631\u0637 \u0627\u0644\u062A\u0627\u0644\u064A: n\u00B2 \u064A\u0643\u062A\u0628 \u0641\u064A \u0634\u0643\u0644 \u0643\u062A\u0627\u0628\u0629 \u0630\u0627 \u0642\u0627\u0639\u062F\u0629 n (\u0639\u0627\u062F\u0629 \u0639\u0634\u0631\u0629) (\u0645\u062B\u0644\u0627:abcd). \u062A\u0642\u0633\u0645 \u0627\u0644\u0643\u062A\u0627\u0628\u0629 \u0625\u0644\u0649 \u062C\u0632\u0623\u064A\u0646 (ab , cd), \u0645\u062C\u0645\u0648\u0639 \u0627\u0644\u062C\u0632\u0623\u064A\u0646 (ab + cd) \u064A\u062C\u0628 \u0623\u0646 \u064A\u0633\u0627\u0648\u064A n \n* \u0645\u062B\u0644\u0627 9 \u0647\u0648 \u0639\u062F\u062F \u0643\u0627\u0628\u0631\u064A\u0643\u0627\u0631 \u0644\u0623\u0646 9\u00B2=81 \u0648 8 + 1 = 9 \n* \u0625\u0630\u0627 \u0643\u0627\u0646 \u0639\u062F\u062F \u0623\u0631\u0642\u0627\u0645 \u0645\u0631\u0628\u0639 \u0627\u0644\u0639\u062F\u062F \u0641\u0631\u062F\u064A\u0627. \u064A\u0643\u0648\u0646 \u0639\u062F\u062F \u0623\u0631\u0642\u0627\u0645 \u0627\u0644\u0646\u0635\u0641 \u0627\u0644\u064A\u0645\u064A\u0646\u064A \u0623\u0643\u0628\u0631 \u0628\u0648\u0627\u062D\u062F \u0645\u0646 \u0627\u0644\u0646\u0635\u0641 \u0627\u0644\u0622\u062E\u0631 \u0645\u062B\u0627\u0644: 9 \u0647\u0648 \u0639\u062F\u062F \u0643\u0627\u0628\u0631\u064A\u0643\u0627\u0631 \u0644\u0623\u0646 297\u00B2=88209 \u0648 88 + 209 = 297 \n* \u0627\u0644\u0623\u0639\u062F\u0627\u062F \u0627\u0644\u0643\u0627\u0628\u0631\u064A\u0643\u0627\u0631 \u0627\u0644\u0623\u0648\u0644\u0649 \u0647\u064A 1,0, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170... \n* \u0628\u0645\u0627 \u0623\u0646 \u062E\u0627\u0635\u064A\u0629 \u0627\u0644\u0639\u062F\u062F \u0643\u0627\u0628\u0631\u064A\u0643\u0627\u0631 \u0645\u062A\u0639\u0644\u0642\u0629 \u0628\u0623\u0631\u0642\u0627\u0645\u0647 \u0641\u0625\u0646 \u0627\u0644\u0623\u0639\u062F\u0627\u062F \u0627\u0644\u0643\u0627\u0628\u0631\u064A\u0643\u0627\u0631 \u062A\u062E\u062A\u0644\u0641 \u0628\u0627\u062E\u062A\u0644\u0627\u0641 \u0646\u0638\u0627\u0645 \u0627\u0644\u0639\u062F."@ar . . "Un nombre de Kaprekar est un entier naturel qui, dans une base donn\u00E9e, lorsqu'il est \u00E9lev\u00E9 au carr\u00E9, peut \u00EAtre s\u00E9par\u00E9 en une partie gauche et une partie droite (non nulle) telles que la somme donne le nombre initial. Exemples en base dix703 est un nombre de Kaprekar car 7032 = 494 209 et 494 + 209 = 703.4 879 est un nombre de Kaprekar car 4 8792 = 23 804 641 et 238 + 04 641 = 4 879. Les nombres de Kaprekar ont \u00E9t\u00E9 principalement \u00E9tudi\u00E9s par le math\u00E9maticien indien D. R. Kaprekar."@fr . . . . . . . . "Nombre de Kaprekar"@ca . "N\u00FAmero de Kaprekar"@pt . . "En matem\u00E1ticas, un n\u00FAmero de Kaprekar (Por: Shri Dattatreya Ramachandra Kaprekar, 1905\u20131986, matem\u00E1tico Indio) es aquel entero no negativo tal que, en una base dada, los d\u00EDgitos de su cuadrado en esa base pueden ser separados en dos n\u00FAmeros que sumados dan el n\u00FAmero original. El ejemplo m\u00E1s simple es 9, su cuadrado es 81 y 8+1= 9.Otro ejemplo es el n\u00FAmero 703, su cuadrado es 494209. Si separamos 494209 en dos nuevos n\u00FAmeros, 494 y 209, obtenemos que 494 + 209 = 703. De igual forma, el n\u00FAmero 297 tambi\u00E9n es un n\u00FAmero de Kaprekar, ya que es posible descomponer el cuadrado 2972 = 88209 en 88 y 209.El segundo n\u00FAmero puede comenzar por cero, pero debe ser positivo. Un ejemplo es 999, ya que 9992=998001 y se descompone en 998 y 001. Por esto mismo, el n\u00FAmero 100 no es un n\u00FAmero de Kaprekar, ya que 100\u00B2=10000 y se descompone en 100 + 00, pero el segundo sumando no es positivo. Matem\u00E1ticamente, sea X un entero no negativo. X es un n\u00FAmero de Kaprekar para la base b si existen n n\u00FAmeros enteros no negativos, A y B, que satisfagan las siguientes condiciones: 0 < B < bnX\u00B2 = Abn + BX = A + B Los primeros n\u00FAmeros de Kaprekar en base 10 son ((sucesi\u00F3n A006886 en OEIS)): 1, 9, 45, 55, 99, 297, 703, ... En binario (base 2) todos los n\u00FAmeros perfectos son n\u00FAmeros de Kaprekar. En cualquier base existen infinitos n\u00FAmeros de Kaprekar, en particular, dada una base b, todos los n\u00FAmeros de la forma bn-1 son n\u00FAmeros de Kaprekar. Los n\u00FAmeros de Kaprekar se nombran en honor a D. R. Kaprekar. No deben confundirse con la constante de Kaprekar, que es el n\u00FAmero 6174."@es . "Let\n\n\n\nThen,\n\n\n\n\nThe two numbers and are\n: \n: \nand their sum is\n\n\n\nThus, is a Kaprekar number."@en . "In mathematics, a natural number in a given number base is a -Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has digits, that add up to the original number. The numbers are named after D. R. Kaprekar."@en . "Nombre de Kaprekar"@fr . "( \uBE44\uC2B7\uD55C \uC774\uB984\uC758 \uCE74\uD504\uB9AC\uCE74 \uC0C1\uC218\uC5D0 \uAD00\uD574\uC11C\uB294 \uD574\uB2F9 \uBB38\uC11C\uB97C \uCC38\uC870\uD558\uC2ED\uC2DC\uC624.) \uC218\uD559\uC5D0\uC11C \uCE74\uD504\uB9AC\uCE74 \uC218(Kaprekar number)\uB294 \uC778\uB3C4\uC758 \uC218\uD559\uC790 \uCE74\uD504\uB9AC\uCE74\uC5D0 \uC758\uD574 \uC815\uC758\uB41C \uC218\uB85C \uC5B4\uB5A4 \uC218\uC758 \uC81C\uACF1\uC218\uB97C \uB450 \uBD80\uBD84\uC73C\uB85C \uB098\uB204\uC5B4 \uB354\uD558\uC600\uC744 \uB54C \uB2E4\uC2DC \uC6D0\uB798\uC758 \uC218\uAC00 \uB418\uB294 \uC218\uB4E4\uC744 \uC758\uBBF8\uD55C\uB2E4. \uC608\uB97C \uB4E4\uC5B4 45\uB294 \uCE74\uD504\uB9AC\uCE74 \uC218\uC778\uB370, 45\u00B2 = 2025\uC774\uACE0, 20+25 = 45\uC774\uAE30 \uB54C\uBB38\uC774\uB2E4."@ko . . "Kaprekar-Zahl"@de . . . "Kaprekartal f\u00F6r en given talbas, \u00E4r ett icke-negativt tal vars kvadrat i den basen kan delas upp i tv\u00E5 delar som summerar till det ursprungliga talet igen. Till exempel \u00E4r 45 ett Kaprekartal eftersom 452 = 2025 och 20 + 25 = 45. Kaprekartal \u00E4r uppkallade efter matematikern D. R. Kaprekar."@sv . . . . "Kaprekartal f\u00F6r en given talbas, \u00E4r ett icke-negativt tal vars kvadrat i den basen kan delas upp i tv\u00E5 delar som summerar till det ursprungliga talet igen. Till exempel \u00E4r 45 ett Kaprekartal eftersom 452 = 2025 och 20 + 25 = 45. Kaprekartal \u00E4r uppkallade efter matematikern D. R. Kaprekar."@sv . "\u0427\u0438\u0441\u043B\u043E \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430 \u0434\u043B\u044F \u0434\u0430\u043D\u043D\u043E\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u044B \u0441\u0447\u0438\u0441\u043B\u0435\u043D\u0438\u044F \u2014 \u044D\u0442\u043E \u043D\u0435\u043E\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u0435 \u0446\u0435\u043B\u043E\u0435 \u0447\u0438\u0441\u043B\u043E, \u043A\u0432\u0430\u0434\u0440\u0430\u0442 \u043A\u043E\u0442\u043E\u0440\u043E\u0433\u043E \u0432 \u044D\u0442\u043E\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u0435 \u043C\u043E\u0436\u043D\u043E \u0440\u0430\u0437\u0431\u0438\u0442\u044C \u043D\u0430 \u0434\u0432\u0435 \u0447\u0430\u0441\u0442\u0438, \u0441\u0443\u043C\u043C\u0430 \u043A\u043E\u0442\u043E\u0440\u044B\u0445 \u0434\u0430\u0451\u0442 \u0438\u0441\u0445\u043E\u0434\u043D\u043E\u0435 \u0447\u0438\u0441\u043B\u043E. \u041D\u0430\u043F\u0440\u0438\u043C\u0435\u0440, 45 \u044F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u0447\u0438\u0441\u043B\u043E\u043C \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430, \u043F\u043E\u0441\u043A\u043E\u043B\u044C\u043A\u0443 452 = 2025 \u0438 20 + 25 = 45. \u0427\u0438\u0441\u043B\u0430 \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430 \u043D\u0430\u0437\u0432\u0430\u043D\u044B \u043F\u043E \u0438\u043C\u0435\u043D\u0438 \u0414. \u0420. \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430."@ru . . "Een Kaprekargetal is in de wiskunde een geheel getal dat de hieronder beschreven eigenschap bezit. De Kaprekargetallen zijn genoemd naar de Indiase wiskundige D.R. Kaprekar (1905\u20131986). Een geheel getal heet, bij een gegeven grondtal, een Kaprekargetal als het kwadraat ervan in twee getallen kan worden gesplitst die bij optelling weer het oorspronkelijke getal geven. Bijvoorbeeld, het 3-cijferige getal 703 is, bij het gebruikelijke grondtal 10, een Kaprekargetal, omdat 7032 = 494209, en 494209 gesplitst kan worden in 494 en 209, en 494 + 209 = 703."@nl . "\u0427\u0438\u0441\u043B\u043E \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430"@ru . . . . "En matem\u00E1ticas, un n\u00FAmero de Kaprekar (Por: Shri Dattatreya Ramachandra Kaprekar, 1905\u20131986, matem\u00E1tico Indio) es aquel entero no negativo tal que, en una base dada, los d\u00EDgitos de su cuadrado en esa base pueden ser separados en dos n\u00FAmeros que sumados dan el n\u00FAmero original. El ejemplo m\u00E1s simple es 9, su cuadrado es 81 y 8+1= 9.Otro ejemplo es el n\u00FAmero 703, su cuadrado es 494209. Si separamos 494209 en dos nuevos n\u00FAmeros, 494 y 209, obtenemos que 494 + 209 = 703. De igual forma, el n\u00FAmero 297 tambi\u00E9n es un n\u00FAmero de Kaprekar, ya que es posible descomponer el cuadrado 2972 = 88209 en 88 y 209.El segundo n\u00FAmero puede comenzar por cero, pero debe ser positivo. Un ejemplo es 999, ya que 9992=998001 y se descompone en 998 y 001. Por esto mismo, el n\u00FAmero 100 no es un n\u00FAmero de Kaprekar, ya q"@es . . . . . "hidden"@en . . . . . . . . . . "Kaprekargetal"@nl . "\u5361\u5E03\u5217\u514B\u6578"@zh . "472129"^^ . "\u0427\u0438\u0441\u043B\u043E \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430 \u0434\u043B\u044F \u0434\u0430\u043D\u043E\u0457 \u0441\u0438\u0441\u0442\u0435\u043C\u0438 \u0447\u0438\u0441\u043B\u0435\u043D\u043D\u044F - \u0446\u0435 \u043D\u0435\u0432\u0456\u0434'\u0454\u043C\u043D\u0435 \u0446\u0456\u043B\u0435 \u0447\u0438\u0441\u043B\u043E, \u043A\u0432\u0430\u0434\u0440\u0430\u0442 \u044F\u043A\u043E\u0433\u043E \u0432 \u0446\u0456\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u0456 \u0447\u0438\u0441\u043B\u0435\u043D\u043D\u044F \u043C\u043E\u0436\u043D\u0430 \u0440\u043E\u0437\u0431\u0438\u0442\u0438 \u043D\u0430 \u0434\u0432\u0456 \u0447\u0430\u0441\u0442\u0438\u043D\u0438, \u0441\u0443\u043C\u0430 \u044F\u043A\u0438\u0445 \u0434\u0430\u0454 \u043F\u043E\u0447\u0430\u0442\u043A\u043E\u0432\u0435 \u0447\u0438\u0441\u043B\u043E. \u041D\u0430\u043F\u0440\u0438\u043A\u043B\u0430\u0434, 45 \u0454 \u0447\u0438\u0441\u043B\u043E\u043C \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430, \u043E\u0441\u043A\u0456\u043B\u044C\u043A\u0438 452 = 2025 \u0456 20 + 25 = 45. \u0427\u0438\u0441\u043B\u0430 \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430 \u043D\u0430\u0437\u0432\u0430\u043D\u0456 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C ."@uk . . "Un nombre de Kaprekar est un entier naturel qui, dans une base donn\u00E9e, lorsqu'il est \u00E9lev\u00E9 au carr\u00E9, peut \u00EAtre s\u00E9par\u00E9 en une partie gauche et une partie droite (non nulle) telles que la somme donne le nombre initial. Exemples en base dix703 est un nombre de Kaprekar car 7032 = 494 209 et 494 + 209 = 703.4 879 est un nombre de Kaprekar car 4 8792 = 23 804 641 et 238 + 04 641 = 4 879. Les nombres de Kaprekar ont \u00E9t\u00E9 principalement \u00E9tudi\u00E9s par le math\u00E9maticien indien D. R. Kaprekar."@fr . . "Proof"@en . . . "\u0427\u0438\u0441\u043B\u043E \u041A\u0430\u043F\u0440\u0435\u043A\u0430\u0440\u0430"@uk . . . . . . . "Een Kaprekargetal is in de wiskunde een geheel getal dat de hieronder beschreven eigenschap bezit. De Kaprekargetallen zijn genoemd naar de Indiase wiskundige D.R. Kaprekar (1905\u20131986). Een geheel getal heet, bij een gegeven grondtal, een Kaprekargetal als het kwadraat ervan in twee getallen kan worden gesplitst die bij optelling weer het oorspronkelijke getal geven. Bijvoorbeeld, het 3-cijferige getal 703 is, bij het gebruikelijke grondtal 10, een Kaprekargetal, omdat 7032 = 494209, en 494209 gesplitst kan worden in 494 en 209, en 494 + 209 = 703."@nl . "In mathematics, a natural number in a given number base is a -Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has digits, that add up to the original number. The numbers are named after D. R. Kaprekar."@en . "( \uBE44\uC2B7\uD55C \uC774\uB984\uC758 \uCE74\uD504\uB9AC\uCE74 \uC0C1\uC218\uC5D0 \uAD00\uD574\uC11C\uB294 \uD574\uB2F9 \uBB38\uC11C\uB97C \uCC38\uC870\uD558\uC2ED\uC2DC\uC624.) \uC218\uD559\uC5D0\uC11C \uCE74\uD504\uB9AC\uCE74 \uC218(Kaprekar number)\uB294 \uC778\uB3C4\uC758 \uC218\uD559\uC790 \uCE74\uD504\uB9AC\uCE74\uC5D0 \uC758\uD574 \uC815\uC758\uB41C \uC218\uB85C \uC5B4\uB5A4 \uC218\uC758 \uC81C\uACF1\uC218\uB97C \uB450 \uBD80\uBD84\uC73C\uB85C \uB098\uB204\uC5B4 \uB354\uD558\uC600\uC744 \uB54C \uB2E4\uC2DC \uC6D0\uB798\uC758 \uC218\uAC00 \uB418\uB294 \uC218\uB4E4\uC744 \uC758\uBBF8\uD55C\uB2E4. \uC608\uB97C \uB4E4\uC5B4 45\uB294 \uCE74\uD504\uB9AC\uCE74 \uC218\uC778\uB370, 45\u00B2 = 2025\uC774\uACE0, 20+25 = 45\uC774\uAE30 \uB54C\uBB38\uC774\uB2E4."@ko . . "\u0639\u062F\u062F \u0643\u0627\u0628\u0631\u064A\u0643\u0627\u0631"@ar . . . . "Numero di Kaprekar"@it . . . "\u5361\u5E03\u5217\u514B\u6578\uFF08Kaprekar number\uFF09\u662F\u5177\u6709\u4EE5\u4E0B\u6027\u8CEA\u7684\u6578\uFF1A \u5C0D\u65BC\u67D0\u500B\u6B63\u6574\u6578\u5728n\u9032\u4F4D\u4E0B\u5B58\u5728\u6B63\u6574\u6578 A, B \u53CA m\uFF0C\u4E14 \n* \n* \n* \u7C21\u55AE\u7684\u8AAA\uFF0C\u82E5\u6B63\u6574\u6578X\u5728n\u9032\u4F4D\u4E0B\u7684\u5E73\u65B9\u53EF\u4EE5\u5206\u5272\u70BA\u4E8C\u500B\u6578\u5B57\uFF0C\u800C\u9019\u4E8C\u500B\u6578\u5B57\u76F8\u52A0\u5F8C\u6070\u7B49\u65BCX\uFF0C\u90A3\u9EBCX\u5C31\u662Fn\u9032\u4F4D\u4E0B\u7684\u5361\u5E03\u5217\u514B\u6578\u3002 \u4F8B\u5982 297 \u5728\u5341\u9032\u4F4D\u4E0B\u662F\u5361\u5E03\u5217\u514B\u6578\uFF0C\u56E0\u70BA\uFF0C\u53EF\u4EE5\u5206\u5272\u6210 88 \u53CA 209\uFF0C\u4E14 88+209=297\u3002\u4E0D\u904E 100 \u5728\u5341\u9032\u4F4D\u4E0B\u4E0D\u662F\u5361\u5E03\u5217\u514B\u6578\uFF0C\u96D6\u7136\uFF0C\u53EF\u4EE5\u5206\u5272\u6210 100 \u53CA 00\uFF0C\u4F46 00 \u4E0D\u662F\u6B63\u6574\u6578\u3002 \u5728\u5341\u9032\u4F4D\u4E0B\uFF0C\u5E7E\u500B\u8F03\u5C0F\u7684\u5361\u5E03\u5217\u514B\u6578\u5982\u4E0B\uFF08OEIS\u6578\u5217\uFF09: 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170 \u5728\u4E8C\u9032\u4F4D\u4E0B\uFF0C\u6240\u6709\u7684\u5B8C\u5168\u6578\u90FD\u662F\u5361\u5E03\u5217\u514B\u6578\u3002 \u6709\u65F6\u5019\uFF0C\u4EBA\u4EEC\u4F1A\u628A6174\u8FD9\u4E2A\u6570\u79F0\u4F5C\u5361\u5E03\u5217\u514B\u6578\uFF0C\u800C\u5176\u5B9E\u8FD9\u662F\u5361\u5E03\u5217\u514B\u5E38\u6570\u3002"@zh . . "En matem\u00E0tiques, els nombres de Kaprekar s\u00F3n nombres naturals que satisfan la condici\u00F3 de que el seu quadrat es pot tallar en dos trossos que, sumats, donen el nombre original. Formalment, doncs, s\u00F3n nombres naturals, , que satisfan les equacions: Aquest nombres foren introduits el 1980 pel matem\u00E0tic indi D. R. Kaprekar. Per exemple, \u00E9s un nombre de Kaprekar perqu\u00E8: i : tamb\u00E9 ho \u00E9s per i . Altres casos son m\u00E9s dif\u00EDcils de trobar: i . A continuaci\u00F3 es mostren altres exemples: La s\u00E8rie OEIS 6886 mostra tots els nombres de Kaprekar en base 10 elevats al quadrat. Aquesta definici\u00F3 es pot generalitzar per a nombres naturals en qualsevol base i elevats a qualsevol pot\u00E8ncia definint una funci\u00F3 de Kaprekar, amb base i pot\u00E8ncia , tal que: , en la qual: i"@ca . .