# Number Theory Division Algorithm interesting problem

1. Dec 24, 2014

### PsychonautQQ

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
None.

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. Dec 24, 2014

### Stephen Tashi

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. Dec 28, 2014

### BvU

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 ?