$$ \begin{array}{r} * \ * \ * \\ \times \ \ \ * \\ \hline 8 \ * \ * \ 1 \\ \hline \end{array} $$ In the above multiplication problem, $*$ is equal to

### Concept & Logic In missing-digit multiplication problems where the same symbol represents the unknown, all instances of that symbol must be the *exact same digit*. We solve this by analyzing unit digit constraints. ### Step-by-Step Solution * **Given:** * A 3-digit number consisting of identical digits ($d d d$) is multiplied by the same digit ($d$). * The product is a 4-digit number starting with $8$, having the same middle digits, and ending in $1$ ($8 d d 1$). * **Calculation / Deduction:** 1. Let the asterisk $*$ be the digit $d$. The equation becomes: $$d d d \times d = 8 d d 1$$ 2. Focus on the unit digit of the multiplication. The unit digit of $d \times d$ must end in $1$. The only single digits that yield a unit digit of $1$ when squared are $1$ ($1 \times 1 = 1$) and $9$ ($9 \times 9 = 81$). 3. Test $d = 1$: $$111 \times 1 = 111$$ This yields a 3-digit number, which contradicts the 4-digit result starting with 8. 4. Test $d = 9$: $$999 \times 9 = 8991$$ This perfectly matches the structure $8 * * 1$, where the middle digits are also $9$. ### Exam Strategy & Shortcut Use the Unit Digit Strategy and Option Elimination. The product ends in $1$. Looking at the options, $1 \times 1 = 1$, $3 \times 3 = 9$, $7 \times 7 = 49$, and $9 \times 9 = 81$. So, the answer must be $1$ or $9$. Since the product starts with $8$, the multiplier must be large. $999 \times 9 = 8991$. Done. ### Common Pitfall Assuming that the asterisks can represent *different* digits is the most common pitfall. If a question asks "$*$ is equal to", it definitively implies a single, uniform value for all asterisks. ### Final Answer Therefore, the correct answer is 9.