1. **Problem statement:** Determine the pairs of integers $(x,y)$ such that the base 5 number $(xy)_5$ equals the base 9 number $(yx)_9$. 2. **Understanding the problem:** - $(xy)_5$ means a two-digit number in base 5 with digits $x$ and $y$. - $(yx)_9$ means a two-digit number in base 9 with digits $y$ and $x$. - We want to find integer digits $x,y$ that satisfy this equality. 3. **Constraints on digits:** - Since digits in base 5 are from 0 to 4, $x,y \in \{0,1,2,3,4\}$. - Since digits in base 9 are from 0 to 8, $x,y \in \{0,1,2,3,4,5,6,7,8\}$. - To satisfy both, $x,y$ must be in $\{0,1,2,3,4\}$. 4. **Express numbers in decimal:** - $(xy)_5 = 5x + y$ - $(yx)_9 = 9y + x$ 5. **Set up the equation:** $$5x + y = 9y + x$$ 6. **Simplify:** $$5x + y = 9y + x$$ $$5x - x = 9y - y$$ $$4x = 8y$$ $$x = 2y$$ 7. **Find integer pairs $(x,y)$ with digits 0 to 4:** - Since $x=2y$ and $x,y \in \{0,1,2,3,4\}$, possible $y$ values are 0,1,2. - For $y=0$, $x=0$. - For $y=1$, $x=2$. - For $y=2$, $x=4$. 8. **Check pairs:** - $(x,y) = (0,0)$: $(00)_5 = 0$, $(00)_9=0$ valid. - $(2,1)$: $(21)_5 = 5*2+1=11$, $(12)_9=9*1+2=11$ valid. - $(4,2)$: $(42)_5=5*4+2=22$, $(24)_9=9*2+4=22$ valid. **Final answer:** The integer pairs $(x,y)$ satisfying $(xy)_5 = (yx)_9$ are: $$\boxed{(0,0), (2,1), (4,2)}$$