Aufgabe:
Seien a, b ∈ {0,1,2,3,4,5,6,7,8,9} beliebig.
Zeige: 7 | ((ab)10 + 2*(abab)10 )
Problem/Ansatz:
Mir fehlt der Ansatz wie ich vorgehen könnte :(
Hallo,
Du könntest alle Möglichkeiten ausprobieren: es sind ja nur 100 ;-)
Schlauer ist folgendes:$$\begin{aligned} (ab)_{10} + 2\cdot(abab)_{10 } &= 10a + b + 2(1010a + 101b) \\ &= 2030a + 203b \\ &= 7\cdot(290a+29b) \end{aligned}$$Man kann also für \(a\) und \(b\) einsetzen, was man will. Der Audruck ist immer durch 7 teilbar.
Gruß Werner
\(\begin{aligned}(00)_{10} + 2\cdot(0000)_{10} &= \phantom{000}0 = 0\cdot 7\\(01)_{10} + 2\cdot(0101)_{10} &= \phantom{0}203 = 29\cdot 7\\(02)_{10} + 2\cdot(0202)_{10} &= \phantom{0}406 = 58\cdot 7\\(03)_{10} + 2\cdot(0303)_{10} &= \phantom{0}609 = 87\cdot 7\\(04)_{10} + 2\cdot(0404)_{10} &= \phantom{0}812 = 116\cdot 7\\(05)_{10} + 2\cdot(0505)_{10} &= 1015 = 145\cdot 7\\(06)_{10} + 2\cdot(0606)_{10} &= 1218 = 174\cdot 7\\(07)_{10} + 2\cdot(0707)_{10} &= 1421 = 203\cdot 7\\(08)_{10} + 2\cdot(0808)_{10} &= 1624 = 232\cdot 7\\(09)_{10} + 2\cdot(0909)_{10} &= 1827 = 261\cdot 7\\(10)_{10} + 2\cdot(1010)_{10} &= 2030 = 290\cdot 7\\(11)_{10} + 2\cdot(1111)_{10} &= 2233 = 319\cdot 7\\(12)_{10} + 2\cdot(1212)_{10} &= 2436 = 348\cdot 7\\(13)_{10} + 2\cdot(1313)_{10} &= 2639 = 377\cdot 7\\(14)_{10} + 2\cdot(1414)_{10} &= 2842 = 406\cdot 7\\(15)_{10} + 2\cdot(1515)_{10} &= 3045 = 435\cdot 7\\(16)_{10} + 2\cdot(1616)_{10} &= 3248 = 464\cdot 7\\(17)_{10} + 2\cdot(1717)_{10} &= 3451 = 493\cdot 7\\(18)_{10} + 2\cdot(1818)_{10} &= 3654 = 522\cdot 7\\(19)_{10} + 2\cdot(1919)_{10} &= 3857 = 551\cdot 7\\(20)_{10} + 2\cdot(2020)_{10} &= 4060 = 580\cdot 7\\(21)_{10} + 2\cdot(2121)_{10} &= 4263 = 609\cdot 7\\(22)_{10} + 2\cdot(2222)_{10} &= 4466 = 638\cdot 7\\(23)_{10} + 2\cdot(2323)_{10} &= 4669 = 667\cdot 7\\(24)_{10} + 2\cdot(2424)_{10} &= 4872 = 696\cdot 7\\(25)_{10} + 2\cdot(2525)_{10} &= 5075 = 725\cdot 7\\(26)_{10} + 2\cdot(2626)_{10} &= 5278 = 754\cdot 7\\(27)_{10} + 2\cdot(2727)_{10} &= 5481 = 783\cdot 7\\(28)_{10} + 2\cdot(2828)_{10} &= 5684 = 812\cdot 7\\(29)_{10} + 2\cdot(2929)_{10} &= 5887 = 841\cdot 7\\(30)_{10} + 2\cdot(3030)_{10} &= 6090 = 870\cdot 7\\(31)_{10} + 2\cdot(3131)_{10} &= 6293 = 899\cdot 7\\(32)_{10} + 2\cdot(3232)_{10} &= 6496 = 928\cdot 7\\(33)_{10} + 2\cdot(3333)_{10} &= 6699 = 957\cdot 7\\(34)_{10} + 2\cdot(3434)_{10} &= 6902 = 986\cdot 7\\(35)_{10} + 2\cdot(3535)_{10} &= 7105 = 1015\cdot 7\\(36)_{10} + 2\cdot(3636)_{10} &= 7308 = 1044\cdot 7\\(37)_{10} + 2\cdot(3737)_{10} &= 7511 = 1073\cdot 7\\(38)_{10} + 2\cdot(3838)_{10} &= 7714 = 1102\cdot 7\\(39)_{10} + 2\cdot(3939)_{10} &= 7917 = 1131\cdot 7\\(40)_{10} + 2\cdot(4040)_{10} &= 8120 = 1160\cdot 7\\(41)_{10} + 2\cdot(4141)_{10} &= 8323 = 1189\cdot 7\\(42)_{10} + 2\cdot(4242)_{10} &= 8526 = 1218\cdot 7\\(43)_{10} + 2\cdot(4343)_{10} &= 8729 = 1247\cdot 7\\(44)_{10} + 2\cdot(4444)_{10} &= 8932 = 1276\cdot 7\\(45)_{10} + 2\cdot(4545)_{10} &= 9135 = 1305\cdot 7\\(46)_{10} + 2\cdot(4646)_{10} &= 9338 = 1334\cdot 7\\(47)_{10} + 2\cdot(4747)_{10} &= 9541 = 1363\cdot 7\\(48)_{10} + 2\cdot(4848)_{10} &= 9744 = 1392\cdot 7\\(49)_{10} + 2\cdot(4949)_{10} &= 9947 = 1421\cdot 7\\(50)_{10} + 2\cdot(5050)_{10} &= 10150 = 1450\cdot 7\\(51)_{10} + 2\cdot(5151)_{10} &= 10353 = 1479\cdot 7\\(52)_{10} + 2\cdot(5252)_{10} &= 10556 = 1508\cdot 7\\(53)_{10} + 2\cdot(5353)_{10} &= 10759 = 1537\cdot 7\\(54)_{10} + 2\cdot(5454)_{10} &= 10962 = 1566\cdot 7\\(55)_{10} + 2\cdot(5555)_{10} &= 11165 = 1595\cdot 7\\(56)_{10} + 2\cdot(5656)_{10} &= 11368 = 1624\cdot 7\\(57)_{10} + 2\cdot(5757)_{10} &= 11571 = 1653\cdot 7\\(58)_{10} + 2\cdot(5858)_{10} &= 11774 = 1682\cdot 7\\(59)_{10} + 2\cdot(5959)_{10} &= 11977 = 1711\cdot 7\\(60)_{10} + 2\cdot(6060)_{10} &= 12180 = 1740\cdot 7\\(61)_{10} + 2\cdot(6161)_{10} &= 12383 = 1769\cdot 7\\(62)_{10} + 2\cdot(6262)_{10} &= 12586 = 1798\cdot 7\\(63)_{10} + 2\cdot(6363)_{10} &= 12789 = 1827\cdot 7\\(64)_{10} + 2\cdot(6464)_{10} &= 12992 = 1856\cdot 7\\(65)_{10} + 2\cdot(6565)_{10} &= 13195 = 1885\cdot 7\\(66)_{10} + 2\cdot(6666)_{10} &= 13398 = 1914\cdot 7\\(67)_{10} + 2\cdot(6767)_{10} &= 13601 = 1943\cdot 7\\(68)_{10} + 2\cdot(6868)_{10} &= 13804 = 1972\cdot 7\\(69)_{10} + 2\cdot(6969)_{10} &= 14007 = 2001\cdot 7\\(70)_{10} + 2\cdot(7070)_{10} &= 14210 = 2030\cdot 7\\(71)_{10} + 2\cdot(7171)_{10} &= 14413 = 2059\cdot 7\\(72)_{10} + 2\cdot(7272)_{10} &= 14616 = 2088\cdot 7\\(73)_{10} + 2\cdot(7373)_{10} &= 14819 = 2117\cdot 7\\(74)_{10} + 2\cdot(7474)_{10} &= 15022 = 2146\cdot 7\\(75)_{10} + 2\cdot(7575)_{10} &= 15225 = 2175\cdot 7\\(76)_{10} + 2\cdot(7676)_{10} &= 15428 = 2204\cdot 7\\(77)_{10} + 2\cdot(7777)_{10} &= 15631 = 2233\cdot 7\\(78)_{10} + 2\cdot(7878)_{10} &= 15834 = 2262\cdot 7\\(79)_{10} + 2\cdot(7979)_{10} &= 16037 = 2291\cdot 7\\(80)_{10} + 2\cdot(8080)_{10} &= 16240 = 2320\cdot 7\\(81)_{10} + 2\cdot(8181)_{10} &= 16443 = 2349\cdot 7\\(82)_{10} + 2\cdot(8282)_{10} &= 16646 = 2378\cdot 7\\(83)_{10} + 2\cdot(8383)_{10} &= 16849 = 2407\cdot 7\\(84)_{10} + 2\cdot(8484)_{10} &= 17052 = 2436\cdot 7\\(85)_{10} + 2\cdot(8585)_{10} &= 17255 = 2465\cdot 7\\(86)_{10} + 2\cdot(8686)_{10} &= 17458 = 2494\cdot 7\\(87)_{10} + 2\cdot(8787)_{10} &= 17661 = 2523\cdot 7\\(88)_{10} + 2\cdot(8888)_{10} &= 17864 = 2552\cdot 7\\(89)_{10} + 2\cdot(8989)_{10} &= 18067 = 2581\cdot 7\\(90)_{10} + 2\cdot(9090)_{10} &= 18270 = 2610\cdot 7\\(91)_{10} + 2\cdot(9191)_{10} &= 18473 = 2639\cdot 7\\(92)_{10} + 2\cdot(9292)_{10} &= 18676 = 2668\cdot 7\\(93)_{10} + 2\cdot(9393)_{10} &= 18879 = 2697\cdot 7\\(94)_{10} + 2\cdot(9494)_{10} &= 19082 = 2726\cdot 7\\(95)_{10} + 2\cdot(9595)_{10} &= 19285 = 2755\cdot 7\\(96)_{10} + 2\cdot(9696)_{10} &= 19488 = 2784\cdot 7\\(97)_{10} + 2\cdot(9797)_{10} &= 19691 = 2813\cdot 7\\(98)_{10} + 2\cdot(9898)_{10} &= 19894 = 2842\cdot 7\\(99)_{10} + 2\cdot(9999)_{10} &= 20097 = 2871\cdot 7\\\end{aligned}\)
Oswald, du bist ja ein richtiger Hobby-Rambo... Brute Force ;)))
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