If $x$ is an even number, then $x^{4n}$, where $n$ is a positive integer, will always have

### Concept & Logic Every non-zero even digit ($2, 4, 6, 8$) has a power cyclicity that results in the unit digit $6$ when raised to the $4$th power (or any multiple of $4$). The only exception is $0$, which always remains $0$. ### Step-by-Step Solution **Given:** $x$ is an even number. We need the unit digit of $x^{4n}$ for any positive integer $n$. **Deduction:** * Since $x$ is an even number, its unit digit can be $0, 2, 4, 6,$ or $8$. * Let's test the $4$th power (setting $n=1$) for each possible even unit digit: * If unit digit is $0$: $0^4 = 0 \rightarrow$ ends in $0$. * If unit digit is $2$: $2^4 = 16 \rightarrow$ ends in $6$. * If unit digit is $4$: $4^4 = 256 \rightarrow$ ends in $6$. * If unit digit is $6$: $6^4 = 1296 \rightarrow$ ends in $6$. * If unit digit is $8$: $8^4 = 4096 \rightarrow$ ends in $6$. * Therefore, the resulting unit digit will always be either $0$ (if $x$ ends in $0$) or $6$ (if $x$ ends in $2, 4, 6,$ or $8$). ### Exam Strategy & Shortcut Instead of writing out all the powers, you can quickly test the two extremes of even numbers. Let $n=1$. Try $x=2$ (gives $2^4 = 16 \rightarrow 6$) and try $x=10$ (gives $10^4 = 10000 \rightarrow 0$). This immediately proves the answer must encompass both $0$ and $6$. ### Common Pitfall The most frequent mistake is forgetting that $0$ is considered an even digit (or that numbers ending in $0$, like $10$, are even). If you only test $2, 4, 6,$ and $8$, you will confidently but incorrectly choose "6 in the unit's place". ### Final Answer **Therefore, the correct answer is either 0 or 6 in the unit's place.**