A number when divided by $6$ leaves remainder $3$. When the square of the same number is divided by $6$, find the remainder.

### Concept & Logic When applying mathematical operations (like squaring, cubing, or multiplying) to a number and then dividing by the *same* divisor, you can bypass the unknown number entirely and apply those exact operations directly to the original remainder to find the new remainder. ### Step-by-Step Solution **Given:** * Original divisor = $6$ * Original remainder = $3$ * New operation = Square the number. * New divisor = $6$ (remains unchanged) **Calculation / Deduction:** * Let the original number be $N$. We know that $N \pmod 6 = 3$. * We need to find the remainder of $N^2$ divided by $6$, which mathematically is $(N^2) \pmod 6$. * According to remainder theorem properties: $$ (N^2) \pmod 6 = (N \pmod 6)^2 \pmod 6 $$ * Substitute the known original remainder into the formula: $$ (3)^2 = 9 $$ * Divide this squared remainder by the divisor ($6$) to normalize it: $$ \frac{9}{6} = 1 \text{ with a remainder of } 3 $$ ### Exam Strategy & Shortcut Skip proving the algebra via variables (like assuming $N = 6k+3$). If a number leaves remainder $R$, its square leaves remainder $R^2$. Mentally calculate $3^2 = 9$, then quickly divide $9$ by $6$ to get the final remainder of $3$. ### Common Pitfall A frequent mistake is assuming the squared remainder ($9$) is the final answer. Students forget that a remainder can NEVER be larger than the divisor ($6$). You must always divide it one last time if it exceeds or equals the divisor. ### Final Answer Therefore, the correct answer is 3.