Find all possible positive integers $n$ such that $3^{n-1} + 5^{n-1} \mid 3^n + 5^n $. Proof explanation Question: Find all possible positive integers $n$ such that $3^{n-1} + 5^{n-1} \mid 3^n + 5^n $.
Solution:  Let's suppose that $3^{n-1}+5^{n-1}\mid 3^n+5^n$, so there is some positive integer $k$ such that $3^n+5^n=k(3^{n-1}+5^{n-1})$. Now, if $k\ge 5$ we have $$k(3^{n-1}+5^{n-1})\ge 5(3^{n-1}+5^{n-1})=5\cdot 3^{n-1}+5\cdot 5^{n-1}>3^n+5^n.$$
This means that $k\le 4$. On the other hand, $3^n+5^n=3\cdot 3^{n-1}+5\cdot 5^{n-1}>3(3^{n-1}+5^{n-1})$, then $k\ge 4$ and thus we deduce that $k=4$. In this case we have $3^n+5^n=4(3^{n-1}+5^{n-1})$, which gives us the equation $5^{n-1}=3^{n-1}$, but if $n>1$ this equation is impossible. 
Hence $n=1$
The above solution is from this answer. Could someone please explain a few things:


*

*why was the value of $k\le4$ and $k\ge4$ picked? where did they get 4?

*why does $3\cdot 3^{n-1}+5\cdot 5^{n-1}>3(3^{n-1}+5^{n-1})$ mean that $k\ge4$ ?

*Why does $5\cdot 3^{n-1}+5\cdot 5^{n-1}>3^n+5^n.$ mean that $k\le4$ ?


Perhaps an easier way to look at it: the equation
$$3^n+5^n=k(3^{n-1}+5^{n-1})$$
can be rewritten
$$5^{n-1}(5-k)=3^{n-1}(k-3)\ .$$
Therefore $5-k$ and $k-3$ must have the same sign.  They can't both be negative as then $k$ would be simultaneously less than $3$ and greater than $5$, and they clearly can't both be zero, so they must both be positive.  Hence
$$k<5\quad\hbox{and}\quad k>3\ ,$$
so $k=4$.  This makes our second equation
$$5^{n-1}=3^{n-1}\ ,$$
and this is true if and only if $n-1=0$.
