Find all distinct positive integers $a,b,c,d,e$ with some given conditions.

Solution 1:

Consider $$2\le a \le b \le c \le d \le e.$$

Denote $$F(a)=\dfrac{a+1}{a},$$ $$F(a,b)=\dfrac{a+1}{a}\cdot \dfrac{b+1}{b},$$ $$...$$ $$F(a,b,c,d,e)=\dfrac{a+1}{a}\cdot \dfrac{b+1}{b}\cdot \dfrac{c+1}{c}\cdot \dfrac{d+1}{d}\cdot \dfrac{e+1}{e}. $$

Note, that $$ (a+1)(b+1)(c+1)(d+1)(e+1)>abcde+1 $$ $$ \Downarrow $$ $$ F(a,b,c,d,e)-\dfrac{1}{abcde}>1\tag{1}. $$

Note, that $$ F(a,b)-\dfrac{1}{ab}>F(a),\\ F(a,b,c)-\dfrac{1}{abc}>F(a,b),\\ F(a,b,c,d)-\dfrac{1}{abcd}>F(a,b,c),\\ F(a,b,c,d,e)-\dfrac{1}{abcde}>F(a,b,c,d).\tag{2} $$

A. To $(a,b,c,d,e)$ be a solution, there must be $$ F(a,b,c,d,e)-\dfrac{1}{abcde}\ge 2\tag{3}, $$ $$ F(a,b,c,d,e)> 2\tag{3'}. $$

If $a\ge 7$, then $$ F(a,b,c,d,e)\le F(a,a,a,a,a)\le\left(\dfrac{8}{7}\right)^5<2, $$ contradiction with $(3')$.

So, if $(a,b,c,d,e)$ is a solution, then $$2\le a \le 6.\tag{4}$$

B. For each possible $a$ consider value $$ N_a = \lfloor F(a)\rfloor $$ and focus on such $b$ only, that $$F(a,b,b,b,b)> N_a+1.$$

For other $b$ one will have

$$ N_a\le F(a)<F(a,b)<F(a,b,c)<F(a,b,c,d)<F(a,b,c,d,e)-\dfrac{1}{abcde}\le F(a,b,b,b,b)-\dfrac{1}{abcde} < N_a+1, $$

i.e. $F(a,b,c,d,e)-\dfrac{1}{abcde}$ is between two consecutive integers, so it cannot be integer.

C. For each possible pair $(a,b)$ consider value $$ N_b = \lfloor F(a,b)\rfloor $$ and focus on such $c$ only, that $$F(a,b,c,c,c)> N_b+1.$$

For other $c$ one will have $$ N_b\le F(a,b)<F(a,b,c)<F(a,b,c,d)<F(a,b,c,d,e)-\dfrac{1}{abcde}\le F(a,b,c,c,c)-\dfrac{1}{abcde} < N_b+1. $$

D. For each possible $(a,b,c)$ consider value $$ N_c = \lfloor F(a,b,c)\rfloor $$ and focus on such $d$ only, that $$F(a,b,c,d,d)> N_c+1.$$

E. For each possible $(a,b,c,d)$ consider value $$ N_d = \lfloor F(a,b,c,d)\rfloor $$ and focus on such $e$ only, that $$F(a,b,c,d,e)> N_d+1.$$

Applying described procedure, one will find four solutions:

$(a,b,c,d,e) = (2,4,16,256,65534)$,
$(a,b,c,d,e) = (2,4,16,284,2506)$,
$(a,b,c,d,e) = (4,4,4,42,5374)$,
$(a,b,c,d,e) = (4,4,8,8,88)$;
two of them have all distinct values $a,b,c,d,e$:

$$(a,b,c,d,e) = (2,4,16,256,65534),$$ $$(a,b,c,d,e) = (2,4,16,284,2506).$$