Proving Congruence Modulo 5 for Powers of n

[knowLittle]

Homework Statement

Suppose n is an integer which is not a divisor of 5.
Prove that $n^{4} \equiv 1 mod5$

The Attempt at a Solution

I know that 16 mod 5 is equivalent to 1 mod5.

$16 = 2^{4}$

2 is not a divisor of 5. How do I prove this for the general case?

I know
$n^{4} -1 = 5k,$ for some k $\in Z$

 

[tiny-tim]

hi knowLittle!

(you mean, not a multiple of 5 ! )

I know
$n^{4} -1 = 5k,$ for some k $\in Z$

if you know that, then that's just another way of saying $n^{4} \equiv 1 mod5$

(otherwise, you know it for n = 2, and obviously for n = 1, so how many more cases do you need to prove?)

 

[knowLittle]

Hi TINY-tim,

The problem states divisor. Also, there's a typo in the $\equiv$ symbol; in the paper it shows '='.

I understand that n=2 and n=1 and satisfies the statement. But, how do I prove this, I am confused as to how to prove this. I assume they want me to prove a general case?

 

[tiny-tim]

hi knowLittle!

I understand that n=2 and n=1 and satisfies the statement. But, how do I prove this, I am confused as to how to prove this. I assume they want me to prove a general case?

yes

i don't know how much you know about mod calculations

since you can prove it for n = 2 (because 16 - 1 = 15), does that help you with any other values of n ?

and since it's obviously true for n = 1, does that help you with any other values of n ?

and are there any other cases left?

(alternatively, do you know Fermat's little theorem?)

 

[knowLittle]

We didn't explicitly covered Fermat's theorem, but I will take a look. I am sure that the solution requires elementary knowledge, for we have not covered much.

I am still lost :/

 

[tiny-tim]

if 2⁴ - 1 is divisble by 5, what can you say about (5n + 2)⁴ - 1 ?

 

[knowLittle]

If 5n+2 is a multiple of 2, then $(5n+2)^{4} -1$ would be divisible by 5. Right?

 

[tiny-tim]

If 5n+2 is a multiple of 2, then $(5n+2)^{4} -1$ would be divisible by 5. Right?

right!

and $(5n+1)^{4} -1$ ?

and $(5n-1)^{4} -1$ ?

and $(5n-2)^{4} -1$ ?

what other cases do you need? ​