5. Let the positive integer $n$ be the sum of the reciprocals of four distinct positive integers $a$, $b$, $c$, $d$. Find the values of $n$ and $a+b+c+d$.


Suppose $1 \leqslant a<b<c<d$.
Then $n=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$
$$
\leqslant 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4} \text {. }
$$

Thus, $n=1$ or 2.
Using the method of magnification and reduction, determine $(a, b, c, d)$ to be
$$
\begin{array}{l}
(2,3,7,42),(2,3,8,24),(2,3,9,18), \\
(2,3,10,15),(2,4,5,20),(2,4,6,12), \\
(1,2,3,6) .
\end{array}
$$

Their sums are $54$, $37$, $32$, $30$, $31$, $24$, $12$.