Which of the following prime numbers while dividing 2176 leaves 9 as remainder?

### Concept & Strategy By the division algorithm, if a number $N$ divided by a divisor $D$ leaves a remainder $R$, then the number $(N - R)$ must be perfectly divisible by $D$. Once we find $(N - R)$, we can test the options to see which prime perfectly divides it. ### Step-by-Step Solution * **Given:** The original number is $2176$ and the target remainder is $9$. * **Step 1: Subtract the remainder from the number.** Perfectly divisible number $= 2176 - 9 = 2167$. The target prime number must be a factor of $2167$. * **Step 2: Check for prime factors using basic divisibility rules.** Instead of testing large options immediately, quickly scan $2167$ for small prime factors. Test for divisibility by $11$: (Sum of digits in odd places) - (Sum of digits in even places). $(2 + 6) - (1 + 7) = 8 - 8 = 0$. Since the result is $0$, $2167$ is perfectly divisible by $11$. * **Step 3: Divide by the found factor to find the paired factor.** $2167 \div 11 = 197$. Thus, $2167 = 11 \times 197$. Both $11$ and $197$ are prime numbers that would leave a remainder of $9$. * **Step 4: Check options.** $197$ is present in the options. ### Exam Strategy & Shortcut Subtract the remainder first to get $2167$. Then, look at the unit digits of the options. * $17 \times (\text{something ending in } 1) \rightarrow 7$ (e.g. $17 \times 100 = 1700$, too small. $17 \times 127$?) * $29 \times (\text{something ending in } 3) \rightarrow 7$ A faster route is running the divisibility rule for $11$ on $2167$. The moment you spot it is divisible by $11$, division yields $197$ directly, matching the last option. ### Common Pitfall Attempting to perform long division of $2176$ by $167$ and $197$ directly without subtracting the remainder first. This makes the arithmetic significantly clunkier. ### Final Answer **Therefore, the correct answer is 197.**