Problema facilmente resolúvel. O programa chama-se "números" e uma vez introduzido um número natural dá ao utilizador a decomposição em factores primos, a função $\varphi$ de Euler, o número de divisores, a soma dos divisores e o produto dos divisores, e como bónus até podia mostrar todos os divisores do número.
O programa foi escrito para me servir de apoio numa disciplina de Teoria dos Números, visto que na altura eu estava com um sério problema de saúde e tinha sérios problemas em concentrar-me (aliás, foi nesse ano em que pela primeira vez tive de desistir numa frequência e deixar para exame).
A função $\varphi$ de Euler, dá, para cada natural $n$ o número de números naturais entre 1 e $n-1$ (inclusive) que é coprimo com $n$, ou, por outras palavras, \[ \varphi(n)= \#\left\{m\in \N_1:m<n \land \left(m \text{ e } n \text{ são primos entre si }\right)\right\}\]
Abaixo vou propor um exercício sem indicar as fórmulas para o resolver, e que é rapidamente resolvido por esse programa de calculadora
Exercício:
Considere o número $n=25401600$. Para este número determine:
- Decomposição de $n$ em factores primos
- número de divisores de $n$
- soma dos divisores de $n$
- produto dos divisores de $n$
- $\varphi(n)$
- Os divisores de $n$
- Decomposição de $25401600$ em factores primos:
$25401600=2^8\times3^4\times5^2\times7^2$
$25401600$ $2$ $12700800$ $2$ $6350400$ $2$ $3175200$ $2$ $1587600$ $2$ $793800$ $2$ $396900$ $2$ $198450$ $2$ $99225$ $3$ $33075$ $3$ $11025$ $3$ $3675$ $3$ $1225$ $5$ $245$ $5$ $49$ $7$ $7$ $7$ $1$
- O número de divisores de $25401600$ é $405$
número de divisores de $25401600$ = $\tau(25401600)=(8+1)\times(4+1)\times(2+1)\times(2+1)=405$
- soma dos divisores de $25401600$: $109255377$
soma dos divisores de $25401600$ = $\displaystyle\frac{2^{8+1}-1}{2-1}\times\displaystyle\frac{3^{4+1}-1}{3-1}\times\displaystyle\frac{5^{2+1}-1}{5-1}\times\displaystyle\frac{7^{2+1}-1}{7-1}=109255377$
- produto dos divisores de $25401600 \approx 3,050473527\times 10^{1499}$:
produto dos divisores de $25401600$ = $25401600^{\frac{\tau(25401600)}{2}}=25401600^{\frac{405}{2}}=\\ 3050473527291822531740041345293141334639821433214236708667640373777187466846318630194\\ 9487298892807258697738201087769997689322881899874797172563978299167031514727744821435\\ 3785218826732663217210386436311097076444607289013988610404724784345668111668186460126\\ 9755883592179122020126123180126841834726986839920892479831881375115138978330538550274\\ 8162513638371583118781073185971083062975611933081110856507983196894999341496394996509\\ 3122171965532585683279551649783233634875314042656032015349875338898739359449069030654\\ 4622884131846965196597429175359622576520634929794994536897916292205148207319758296280\\ 8761845614002132058673973259309044737132095162540601973927995102974372145280524186218\\ 4767447352454155282281847316301542497360257699942103612325756704426945835441772896564\\ 4149847241035507859817472009316357318712266669730563110807016171044701603460590155380\\ 6797553649708463435983769227152753101366622137813294565298246991679701906184276978850\\ 6889738588331844184392532814011815755326497291985162421953577246183809913211380900800\\ 4097694439501323737492915590833287001263683936187738812398146247639422730240000000000\\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000\\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000\\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000\\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000\\ 0000000000000000000000000000000000000000000000000000000
\approx 3,050473527\times 10^{1499}$
- $\varphi(25401600)=5806080$
$\varphi(25401600)=2^{8-1}(2-1)\times3^{4-1}(3-1)\times5^{2-1}(5-1)\times7^{2-1}(7-1)=5806080$
- $D_{25401600}=\left\{\right.$ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 126, 128, 135, 140, 144, 147, 150, 160, 162, 168, 175, 180, 189, 192, 196, 200, 210, 216, 224, 225, 240, 245, 252, 256, 270, 280, 288, 294, 300, 315, 320,324, 336, 350, 360, 378, 384, 392, 400, 405, 420, 432, 441, 448, 450, 480, 490, 504, 525, 540, 560, 567, 576, 588, 600, 630, 640, 648, 672, 675, 700, 720, 735, 756, 768, 784, 800, 810, 840, 864, 882, 896, 900, 945, 960, 980, 1008, 1050, 1080, 1120, 1134, 1152, 1176, 1200, 1225, 1260, 1280, 1296, 1323, 1344, 1350, 1400, 1440, 1470, 1512, 1568, 1575, 1600, 1620, 1680, 1728, 1764,1792, 1800, 1890, 1920, 1960, 2016, 2025, 2100, 2160, 2205, 2240, 2268, 2304, 2352, 2400, 2450, 2520, 2592, 2646, 2688, 2700,2800, 2835, 2880, 2940, 3024, 3136, 3150, 3200, 3240, 3360, 3456, 3528, 3600, 3675, 3780, 3840, 3920, 3969, 4032, 4050, 4200, 4320, 4410, 4480, 4536, 4704, 4725, 4800, 4900, 5040, 5184, 5292, 5376, 5400, 5600, 5670, 5760, 5880, 6048, 6272, 6300, 6400,6480, 6615, 6720, 6912, 7056, 7200, 7350, 7560, 7840, 7938, 8064, 8100, 8400, 8640, 8820, 8960, 9072, 9408, 9450, 9600, 9800,10080, 10368, 10584, 10800, 11025, 11200, 11340, 11520, 11760, 12096, 12544, 12600, 12960, 13230, 13440, 14112, 14175, 14400, 14700, 15120, 15680, 15876, 16128, 16200, 16800, 17280, 17640, 18144, 18816, 18900, 19200, 19600, 19845, 20160, 20736, 21168, 21600, 22050, 22400, 22680, 23520, 24192, 25200, 25920, 26460, 26880, 28224, 28350, 28800, 29400, 30240, 31360, 31752, 32400, 33075, 33600, 34560, 35280, 36288, 37632, 37800, 39200, 39690, 40320, 42336, 43200, 44100, 44800, 45360, 47040, 48384, 50400, 51840, 52920, 56448, 56700, 57600, 58800, 60480, 62720, 63504, 64800, 66150, 67200, 70560, 72576, 75600, 78400, 79380, 80640, 84672, 86400, 88200, 90720, 94080, 99225, 100800, 103680, 105840, 112896, 113400, 117600, 120960, 127008, 129600, 132300, 134400, 141120, 145152, 151200, 156800, 158760, 169344, 172800, 176400, 181440, 188160, 198450, 201600, 211680, 226800, 235200, 241920, 254016, 259200, 264600, 282240, 302400, 313600, 317520, 338688, 352800, 362880, 396900, 403200, 423360, 453600, 470400, 508032, 518400, 529200, 564480, 604800, 635040, 705600, 725760, 793800, 846720, 907200, 940800, 1016064, 1058400, 1209600, 1270080, 1411200, 1587600, 1693440, 1814400, 2116800, 2540160, 2822400, 3175200, 3628800, 4233600, 5080320, 6350400, 8467200, 12700800, 25401600 $\left.\right\}$
Programas de calculadora: (.g1m - Modelos Casio fx-9860GII e fx9750GII; .g3m - Modelos Casio fx-cg10 fx-cg20 e fx-cg50; .8xp - Modelos Texas Instruments TI-84Plus CE e CET, .tns - Modelos Texas Instruments nSpire CX e nSpire CX CAS)
.g1m .g3m .8xp .tns 
- soma dos divisores de $25401600$: $109255377$
