A number when divided by 5 leaves the remainder 3. What is the remainder when the square of the same number is divided by 5?

### Concept & Logic According to the properties of remainders in division, whatever operation is performed on the original number (addition, multiplication, squaring) can be directly applied to its remainder to find the new remainder. $$ \text{New Remainder} = (R_1)^2 \pmod{D} $$ ### Step-by-Step Solution * Let the unknown number be $N$. * When $N$ is divided by 5, the remainder is 3. We can write this algebraically as: $N = 5k + 3$ (where $k$ is an integer quotient) * The question asks for the remainder when the square of the number ($N^2$) is divided by 5. Let's square our equation: $N^2 = (5k + 3)^2$ $N^2 = 25k^2 + 30k + 9$ * Now, divide this entire expression by 5. * The terms $25k^2$ and $30k$ are completely divisible by 5, leaving no remainder. * We only need to find the remainder when 9 is divided by 5. * $9 \div 5$ gives a quotient of 1 and a remainder of 4. ### Exam Strategy & Shortcut Bypass the algebra entirely. When a problem asks for the remainder of a squared number, simply square the original remainder and divide that by the divisor. Original remainder = 3. Square it = $3^2 = 9$. Divide 9 by 5. The remainder is 4. This takes less than 3 seconds. ### Common Pitfall A frequent mistake is squaring the remainder (getting 9) and selecting it as the answer, forgetting that a remainder can never be larger than the divisor (which is 5). You must perform that final division step. ### Final Answer Therefore, the correct answer is **4**.