elementary number theory - If $gcd(m,15)=gcd(n,15)=1,$ show that $15mid (m^4+n^4)$ or $15mid (m^4-n^4)$.

If $\gcd(m,15)=\gcd(n,15)=1$ show $15\mid (m^4+n^4)$ or $15\mid (m^4-n^4)$.

This is what I have so far but I'm stuck on essentially the last step.

Proof: Assume $\gcd(m,15)=\gcd(n,15)=1$. By the Euler's Phi Function: If $\gcd(a,m)=1$ then $a^{\phi (m)}\equiv 1 \pmod m$.

$\phi(15)=\phi(5)\phi(3)=(5-1)(3-1)=8$

Hence, $m^8\equiv 1 \pmod{15}$ and $n^8\equiv 1 \pmod{15}$.

Then, $m^8\equiv 1 \pmod{15}\iff 15\mid m^8-1$. Similarly $15\mid n^8-1$.

Divisibility property sates if $a\mid b$ and $a\mid c$ then $a\mid b\pm c$.

So $15\mid (m^8-1)-(n^8-1) \Rightarrow 15\mid m^8-n^8$

So I believe I'm right up to this point, please correct if I'm not. I know that $m^8-n^8$ factors to $(m^4+n^4)(m^4-n^4)$ however since $15$ is not a prime number I can't assume $15\mid m^4+n^4$ or $15\mid m^4-n^4$.

Please help on these last few steps. Thanks.

Answer

Since none of the existing answers is built-up on OP's work, I posted another one.

OP has proved $15\mid m^8-n^8 = (m^4-n^4)(m^4+n^4)$.

Since $15$ is a composite number, to apply Euclid's Lemma, we have to do so in two steps (on prime number $3$ and $5$).

Note that (without Fermat's Little Theorem)

• $a^2 \equiv 0, 1 \pmod 3$, so $a^4 \equiv 0,1 \pmod 3$.
  
  
  
  
  
  • If $3 \mid m^4 + n^4$, it's easy to see that the only possibility is that $3 \mid m,n$, which contradicts $\gcd(m,15) = \gcd(n,15) = 1$
  
  
  • Use Euclid's Lemma to show that $3 \mid m^4 - n^4$
  
  


• $a^2 \equiv 0, 1, 4 \pmod 5$, so $a^4 \equiv 0,1 \pmod 5$, and $m^4 + n^4 \equiv 0,1,2 \pmod 5$.
  
  
  
  
  • If $5 \mid m^4 + n^4$, its again easy to see that the only possibility is $5 \mid m,n$, which contradicts $\gcd(m,15) = \gcd(n,15) = 1$
  
  
  • Use Euclid's Lemma to show that $5 \mid m^4 - n^4$
  
  

Since $\gcd(3,5) = 1$, $15 \mid m^4 - n^4$.