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Number Theory Division Algorithm interesting problem

  1. Dec 24, 2014 #1
    1. The problem statement, all variables and given/known data
    Not actually for homework, but i didn't know where to post this.

    Problem: Show that any integer to the fourth power can be expressed as either 5k or 5k+1 where k is an integer.

    2. Relevant equations

    3. The attempt at a solution
    My starting point is to consider that all integers can be expressed as either:
    2x or 2x+1

    taking these to the fourth power I arrive at:
    16k or 16k + 1

    now i'm stuck, any tips? am i even on the right trail here?
  2. jcsd
  3. Dec 24, 2014 #2

    Stephen Tashi

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    Science Advisor

    You could start by considering they can be expressed as one of:
    5x, 5x+1, 5x+2, 5x+3, 5x+4

    If you are familiar with arithmetic modulo 5, you could compute the 4th power of each of the 5 residue classes.
  4. Dec 28, 2014 #3


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    2017 Award

    2x4 = 16k I can understand.
    But perhaps you want to reconsider (2x+1)4 = 16k+1. How did you do that ?

    On another note: are you familiar with proof by induction ?
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