Lesson Find the last three digits of these numbers

Find the last three digits of these numbers

Problem 1

1.   = .

    Let us go up on this upstairs from the bottom to the top step by step.



2.  First consider the number  .  It is  19683:

     = 19683.

     Write it in the form   = .


     It is clear that for    and  for    the last three digits are determined by last 
     
     three digits "683" of the number  .

     The part    does not affect the last three digits of the number  .

     It is exactly what the binomial theorem says and provides in this situation.


     Therefore, in finding the three last digits of the number    we can track only  for    and do not concern about other terms.


     It implies that the last three digits of the number    are exactly the same as the last three digits of the number  .


     The number   = 318611987,  as easy to calculate (I used Excel in my computer),  so its last three digits are  987.



3.  Now we can make the next (and the last) step up on this upstairs in the same way.

    The last three digits of the number    are the same as the last three digits of the number  ,  and it is easy to calculate.

     = 961504803,    and its last three digits are  803.


    Therefore, the last three digits of the number    are 803.

Problem 2

    First consider the number  .  It is  15876.

    Next                       .  It is  2000376.

    Next                       .  It is  252047376.


    Notice that    and    have the last 3 digits  376.


    Now it is clear that for    the last three digits are determined by last three digits  of the number  .


    To prove it, write    in the form

        = 1000*N + 376,


    and notice that  376*126 = 47376  has the three last digits  376.


    It implies, by the method of Mathematical induction, that ALL  the numbers    have the three lest numbers 376, starting from n = 3.