In the product $459 \times 46 \times 28x \times 484$, the digit in the unit place is $8$. The digit to come in place of $x$ is

### Concept & Logic When an unknown digit $x$ is part of a product, we can isolate it by finding the unit digit of all known components. We then set up a modular equation to find which value of $x$ produces the target final unit digit. ### Step-by-Step Solution **Given:** The product is $459 \times 46 \times 28x \times 484$, and its final unit digit is $8$. Let the unit digit of $28x$ be $x$. **Calculation:** * Extract the unit digits of the known numbers: $9$, $6$, $x$, and $4$. * Set up the equation for the product of these unit digits: $$9 \times 6 \times x \times 4 \equiv 8 \pmod{10}$$ * Multiply the known digits, simplifying to single digits at each step: * $9 \times 6 = 54 \rightarrow$ Unit digit is $4$. * $4 \times 4 = 16 \rightarrow$ Unit digit is $6$. * Now, substitute this back into the equation: $$6 \times x \equiv 8 \pmod{10}$$ * This means we need a digit $x$ such that $6 \times x$ ends in $8$. * Test the digits from $0$ to $9$: * $6 \times 3 = 18$ (Ends in $8$) * $6 \times 8 = 48$ (Ends in $8$) * So, $x$ can be either $3$ or $8$. * Looking at the given options, $3$ is present while $8$ is not. ### Exam Strategy & Shortcut Instead of mentally testing all numbers from $0$ to $9$, plug the given multiple-choice options directly into the simplified equation $6 \times x$. Option (a) is $3 \rightarrow 6 \times 3 = 18$ (Matches). Option (b) is $5 \rightarrow 6 \times 5 = 30$ (Fails). Option (c) is $7 \rightarrow 6 \times 7 = 42$ (Fails). Option elimination is the fastest path here. ### Common Pitfall Students often forget that there can be multiple valid digits that satisfy the unit digit equation (like $3$ and $8$ in this case). Don't panic if your first mental answer isn't in the options; check for the alternate solution. ### Final Answer **Therefore, the correct answer is 3.**