Exponentiation mod n - execution of the program

In my previous post I detailed a program for finding remainders when a number like 67¹¹ is divided by 41. In my first post I showed that 67¹¹ is the 21 digit number 122130132904968017083 and the remainder on dividing by 41 is 12. On executing the program the display shows X? Enter 67 and press [EXE]. Next A? is displayed and this is the index, so enter 11 and press [EXE]. Then enter the divisor 41 when B? is displayed. On pressing [EXE] again the calculator computes the remainder D and this is shown as 12, which is as expected. Pressing [EXE] again allows another set of numbers to be entered.

If we didn't know what the number 67¹¹ looks like we could at least determine what its last two digits were by replacing the divisor B in the above with 100. When we do this the program calculates the remainder as 83 as expected.

The program is surprisingly fast even for what are very large numbers. For example 67⁶¹⁷ is the 1,127 digit number (from WolframAlpha)

48770057363439563967134348355325038065481286153662007642823470550486

71980560027835524800048781488164580806474123084454431777128171666866

18607377904496236891798400886042117666614291800426648265371610864835

40929572898386140596732260223574328368309073158031647754279482918284

65663724881612657390133487402742747112192939724971784183405095503257

69918877553469511522677242675385108662437741711997020059606722804142

09675268052772569724085754968798539106891583795237329222561781805522

85062589820240643630480393850097329533051647302120862011637881575053

71309698184194685080288379701970330799370917504675997998122640686686

98423317552437389277363219335647530610628399241914162225364781445082

70644504402991235669883493855155234017024331172877674569648340943127

70360444117552920156878524196353157664201104105212692362637491637938

54079014668035647988158392840059044333758548155163870985659977545610

52000559917423988628809593315580421167977682921477636773152575740312

22156667453352073011960706603866023551139632454252812187342672444656

55949111316392732278972914732349074457742403565481714331119758806068

559399641911405354923544154126627269027

However, using the fx-50F program we can determine that the last digit of this number is 7 in only a few seconds (this is 67⁶¹⁷mod 10).