<s>[INST] $n \in \mathbb{N} \ 5|\ 2^{2n+1}+3^{2n+1}$ show for all $n \in \mathbb{N}$, 
$$5|\ 2^{2n+1}+3^{2n+1}$$
Indeed,
we've to show that : $2^{2n+1}+3^{2n+1}=0[5] $
note that $2^{2n+1}+3^{2n+1}=2.4^n+3.9^n= $
 [/INST] You can do it by induction. For $n=0$, we have 
$$2^{2n+1}+3^{2n+1}=2+3=5$$
which is divisible by $5$. Now suppose that 
$2^{2k+1}+3^{2k+1}$ is divisible by $5$, i.e. 
$$\tag{1}2^{2k+1}+3^{2k+1}=5N$$
for some integer $N$. Now consider 
$$2^{2(k+1)+1}+3^{2(k+1)+1}=4\cdot 2^{k+1}+9\cdot3^{2k+1}=
4\cdot (2^{k+1}+3^{2k+1})+5\cdot 3^{2k+1}=5(N+3^{2k+1})$$
where the last equality follows from $(1)$, which is divisible by $5$.
</s>