When $n$ is divided by 4, the remainder is 3. What is the remainder when $2n$ is divided by 4?

### Concept & Logic Similar to squaring, if a number is multiplied by a constant, its remainder when divided by the same divisor is also multiplied by that constant. $$ \text{New Remainder} = (2 \times R_1) \pmod{D} $$ ### Step-by-Step Solution * **Algebraic Method:** * Let the number be $n$. * According to the division rule, we can express $n$ as: $n = 4k + 3$ (where $k$ is some integer) * The question asks about $2n$. Multiply the entire equation by 2: $2n = 2(4k + 3)$ $2n = 8k + 6$ * Now, we divide $2n$ by 4 to find the remainder. * The term $8k$ is perfectly divisible by 4. * We only evaluate the constant term, 6, divided by 4. * $6 \div 4$ yields a quotient of 1 with a remainder of 2. ### Exam Strategy & Shortcut The absolute fastest method for variable remainder problems is to assume the smallest possible valid integer for the variable. If $n$ divided by 4 leaves a remainder of 3, assume the quotient is 0. Thus, $n = 3$. The problem asks for $2n$. Double it: $2 \times 3 = 6$. Now divide 6 by 4. The remainder is 2. ### Common Pitfall Multiplying the remainder ($3 \times 2 = 6$) and marking 6 as the answer (Option d) is a very common trap. A remainder can never equal or exceed its divisor (4). Always complete the final modular reduction step. ### Final Answer Therefore, the correct answer is **2**.