SOLUTION: Find the last four(4) digits of 1444^144^4.

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Question 1109365: Find the last four(4) digits of 1444^144^4.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13216) About Me  (Show Source):
You can put this solution on YOUR website!


There might be a relatively easy algebraic method for doing this.... But a spreadsheet can help you find the answer with little effort.

Use the spreadsheet to find the pattern of numbers 1444^n, mod 10000. It turns out the pattern repeats in a cycle of length 250.

Use a spreadsheet or another mathematical tool to find that 144^4, mod 250 is equal to 196.

Then look in your spreadsheet to find the last 4 digits of 1444^196 are 2656.

Answer: 2656

Answer by ikleyn(52914) About Me  (Show Source):
You can put this solution on YOUR website!
.
I just saw several (more than one) posts in this forum asking about last four digits of the numbers of the form  N%5En.
Probably,  it is good time to shed more light on this subject. 

    Let  N  be an ARBITRARY fixed natural number (positive integer), and m be another fixed positive integer number.

    Consider infinite sequence  N%5En,  n = 1, 2, 3, 4, . . . 

    Then starting from some  N%5Ek, the  "m"  last digits of  the numbers  N%5En will repeat cyclically.

This statement seems to be very advanced,  but its proof is in couple of lines. 

    For the sequence N%5En,  the sequence of its "m last digits" is the sequence  N%5En mod 10%5Em.  

    There is only finite number of different m-digit numbers,  so it will happen INEVITABLY  N%5En mod 10%5Em = N%5Ek mod 10%5Em  for 
    some  n > k  for the first time  (the Dirichlet's principle,  or so name "pigeons principle").


    Then after that  this equality will repeat periodically/cyclically.