4. Zadatak

Označimo zapisane brojeve $a$, $b$, $c$, $d$, $e$. Tada je

$$\begin{aligned} abcd&=9,\\ abce&=12,\\ abde&=18,\\ acde&=24,\\ bcde&=36. \tag{*} \end{aligned}$$

Pomnožimo li ovih pet jednadžbi, dobivamo

$$a^4\cdot b^4\cdot c^4\cdot d^4\cdot e^4=9\cdot12\cdot18\cdot24\cdot36,$$ $$(abcde)^4=3^2\cdot3\cdot2^2\cdot2\cdot3^2\cdot3\cdot2^3\cdot2^2\cdot3^2=2^8\cdot3^8,$$ $$abcde=\pm4\cdot9=\pm36. \tag{**}$$

Ako je $abcde=36$, uspoređivanjem $(*)$ i $(**)$ dobivamo

$$abcde=9e=36\quad\Longrightarrow\quad e=4,$$

a analogno se dobije i

$$d=\frac{36}{12}=3,\qquad c=\frac{36}{18}=2,\qquad b=\frac{36}{24}=\frac32,\qquad a=\frac{36}{36}=1.$$

Tako dobivamo skup $\left\{1,\dfrac32,2,3,4\right\}$.

Ako je $abcde=-36$, analogno dobivamo

$$e=-4,\qquad d=-3,\qquad c=-2,\qquad b=-\frac32,\qquad a=-1.$$

Traženi skupovi brojeva su

$$\left\{1,\frac32,2,3,4\right\} \qquad\text{i}\qquad \left\{-1,-\frac32,-2,-3,-4\right\}.$$