A number when divided by $3$ leaves a remainder $1$. When the quotient is divided by $2$, it leaves a remainder $1$. What will be the remainder when the number is divided by $6$?

### Concept & Logic This is a classic **Successive Division** problem disguised with slightly different phrasing ("When the quotient is divided by..."). We find the smallest possible original number by working backward and then apply the final division condition. ### Step-by-Step Solution * **Given:** * First divisor = $3$, Remainder = $1$. * Second divisor = $2$, Remainder = $1$. * **Deduction:** * We work backward starting from the second division. To find the smallest valid number, assume the final quotient is $0$. * Let the intermediate quotient be $q$. $$q = (2 \times 0) + 1 = 1$$ * Now, reconstruct the original number $N$ using the first divisor and remainder, along with our intermediate quotient $q$. $$N = (3 \times q) + 1$$ $$N = (3 \times 1) + 1 = 4$$ * **Calculation:** * We have determined the original number $N$ is $4$. * The question asks for the remainder when $N$ is divided by $6$. * $$4 \div 6 = 0$$ with a remainder of $4$. ### Exam Strategy & Shortcut You can verify this quickly by assuming the final quotient is $1$ instead of $0$. $q = (2 \times 1) + 1 = 3$. $N = (3 \times 3) + 1 = 10$. Now divide $10$ by $6$. $10 \div 6 \rightarrow \text{Remainder } 4$. The remainder remains consistent regardless of the final quotient you choose. Always pick $0$ for the fastest mental math. ### Common Pitfall A frequent trap is reading this as simultaneous independent conditions (e.g., $N \pmod 3 = 1$ and $N \pmod 2 = 1$). That would give $N=7$, which leaves a remainder of $1$ when divided by $6$ (an option not present, luckily). Pay close attention to the phrase "When the **quotient** is divided...", indicating successive division. ### Final Answer **Therefore, the correct answer is 4.**