When a number is divided by 13, the remainder is 11. When the same number is divided by 17, the remainder is 9. What is the number?

### Concept & Strategy This problem represents a system of linear congruences. While there are formal algebraic methods to solve this (like the Chinese Remainder Theorem), they are far too time-consuming for competitive exams. The optimal strategy is direct option elimination. ### Step-by-Step Solution * Let the unknown number be $N$. We have two conditions: 1) $N \div 13 \rightarrow \text{Remainder } 11$ 2) $N \div 17 \rightarrow \text{Remainder } 9$ * **Test Option (a) 339:** * Divide by 13: $339 = 13 \times 26 + 1$. Remainder is 1 (Fails condition 1). * **Test Option (b) 349:** * Divide by 13: $349 = 13 \times 26 + 11$. Remainder is 11 (Passes condition 1). * Divide by 17: $349 = 17 \times 20 + 9$. Remainder is 9 (Passes condition 2). * Since 349 perfectly satisfies both conditions, we can stop testing here. ### Exam Strategy & Shortcut Use mental estimation to speed up the division checks. For 13: You know $13 \times 10 = 130$, so $13 \times 20 = 260$. $349 - 260 = 89$. The closest multiple of 13 to 89 is $13 \times 6 = 78$. $89 - 78 = 11$. Condition 1 confirmed. For 17: You know $17 \times 20 = 340$. $349 - 340 = 9$. Condition 2 confirmed instantly. ### Common Pitfall Students often fall into the trap of trying to build and solve a general algebraic formula like $N = 13x + 11 = 17y + 9$. This turns into a tedious Diophantine equation and drains precious minutes. When the options contain specific numbers, always use them to back-solve. ### Final Answer Therefore, the correct answer is **349**.