After the division of a number successively by $3$, $4$ and $7$, the remainders obtained are $2$, $1$ and $4$ respectively. What will be the remainder if $84$ divides the same number?

### Concept & Logic This problem is based on the concept of **Successive Division**. In successive division, the quotient of the first division is taken as the dividend for the second division, and so on. The general relation is: $$\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$$ ### Step-by-Step Solution * **Given:** * Divisors in sequence: $3, 4, 7$ * Respective remainders: $2, 1, 4$ * **Deduction (Working Backwards):** * Let's assume the final quotient for the last division (by $7$) is $0$ to find the smallest possible valid number. * **Step 1:** The number before being divided by $7$ (let's call it quotient $q_2$). $$q_2 = (7 \times 0) + 4 = 4$$ * **Step 2:** The number before being divided by $4$ (let's call it quotient $q_1$). $$q_1 = (4 \times 4) + 1 = 16 + 1 = 17$$ * **Step 3:** The original number $N$ before being divided by $3$. $$N = (3 \times 17) + 2 = 51 + 2 = 53$$ * **Calculation:** * The smallest possible original number $N$ is $53$. * The question asks for the remainder when this number is divided by $84$. * Since $53$ is less than $84$, dividing $53$ by $84$ simply leaves a remainder of $53$. ### Exam Strategy & Shortcut Whenever you are asked to divide the number by the product of the successive divisors (here, $3 \times 4 \times 7 = 84$), the smallest number you derive by assuming the final quotient as $0$ will **always** be the remainder itself. Just work backwards once: $7(0)+4=4$, $4(4)+1=17$, $3(17)+2=53$. Mark $53$ and move on. ### Common Pitfall A major mistake is treating "successive division" like normal division (trying to find an LCM or solving simultaneous equations). Always remember the chain reaction: the quotient of one step is the dividend of the next. ### Final Answer **Therefore, the correct answer is 53.**