Let $N -$ number of fives $(a_1, a_2, a_3, a_4, a_5)$ positive integers, satisfying the condition $$\frac {1}{a_1}+\frac {1}{a_2}+\frac {1}{a_3}+\frac {1}{a_4}+\frac {1}{a_5}=1.$$ Find out even or odd number is $N$.
My work so far:
$$\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}=1 -$$ first five $\left({5};{5};{5};{5};{5} \right)$ $$\frac{1}{2}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=1 -$$ the following five fives $\left(2;8;{8};{8};{8} \right)$; $\left({8};{2};{8};{8};{8} \right)$;...; $\left({8};{8};{8};{8};{2} \right)$ $$\frac{1}{3}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=1 -$$ the following five fives $\left({3};{6};{6};{6};{6} \right)$; $\left({6};{3};{6};{6};{6} \right)$;...; $\left({6};{6};{6};{6};{3} \right)$. $$N=1+5+5+k?$$ $$k=?$$
HINT
You just have to determine if $N$ is even or odd.
If $a_1, a_2, a_3, a_4, a_5$ are all distinct, there are $120$ ways to arrange them.
Similarly, if two of them were the same there are $60$ ways to arrange them.
In both cases, there are an even number of ways to arrange them.
However, these cases, which are part of $N$ will make no difference in whether of not $N$ is even or odd.
Note the only cases where there are an odd number of ways to arrange them is when at least $4$ numbers are the same.
I think you can continue from here.