Number Theory Division Algorithm interesting problem

  • #1
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11

Homework Statement


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.

Homework Equations


None.

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?
 

Answers and Replies

  • #2
Stephen Tashi
Science Advisor
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My starting point is to consider that all integers can be expressed as either:
2x or 2x+1
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.
 
  • #3
BvU
Science Advisor
Homework Helper
14,126
3,546
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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