If $m$ and $n$ are positive integers, then the digit in the unit's place of $5^n + 6^m$ is always

### Concept & Rule The numbers $5$ and $6$ are special because they are invariant under exponentiation. Any positive integer power of $5$ will end in $5$, and any positive integer power of $6$ will end in $6$. ### Step-by-Step Solution **Given:** Find the unit digit of $5^n + 6^m$ where $m, n > 0$. **Calculation:** * Analyze the first term, $5^n$: * $5^1 = 5$, $5^2 = 25$, $5^3 = 125\dots$ The unit digit is always $5$. * Analyze the second term, $6^m$: * $6^1 = 6$, $6^2 = 36$, $6^3 = 216\dots$ The unit digit is always $6$. * Add the unit digits together: $$5 + 6 = 11$$ * The unit digit of this sum ($11$) is $1$. ### Exam Strategy & Shortcut This is a pure "knowledge check" question that requires zero arithmetic. Commit the static digits ($0, 1, 5, 6$) to memory. Seeing $5^n + 6^m$ should instantly trigger $5 + 6 = 11 \rightarrow 1$. This should take no more than 3 seconds on an exam. ### Common Pitfall Students sometimes get intimidated by abstract variables like $m$ and $n$ with unknown values, thinking they need to set up algebraic equations or that the answer depends on whether $m$ and $n$ are odd or even. Recognize when variables are irrelevant decoys. ### Final Answer **Therefore, the correct answer is 1.**