A number being successively divided by $3$, $5$ and $8$ leaves remainders $1$, $4$ and $7$ respectively. Find the respective remainders if the order of divisors be reversed.

### Concept & Strategy Successive division means that the quotient of the first division becomes the dividend for the second division, and so forth. To find the original number, you must work backward from the final quotient (which we assume is $1$ to find the smallest valid number) using the formula: $\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$. ### Step-by-Step Solution **Given:** * Successive divisors: $3$, $5$, $8$ * Successive remainders: $1$, $4$, $7$ **Calculation / Deduction:** * Let the original number be $x$, and the subsequent quotients be $y$ and $z$. * **Step 1: Reconstruct the number from the bottom up.** * Assume the final quotient after dividing by $8$ is $1$. * Find $z$ (dividend for the 3rd division): $$ z = (8 \times 1) + 7 = 15 $$ * Find $y$ (dividend for the 2nd division): $$ y = (5 \times z) + 4 = (5 \times 15) + 4 = 79 $$ * Find $x$ (the original number): $$ x = (3 \times y) + 1 = (3 \times 79) + 1 = 238 $$ * **Step 2: Reverse the divisors ($8$, $5$, $3$) and perform successive division on $238$.** * $238 \div 8$: Quotient is $29$, Remainder is $6$. * $29 \div 5$: Quotient is $5$, Remainder is $4$. * $5 \div 3$: Quotient is $1$, Remainder is $2$. * The respective remainders are $6$, $4$, and $2$. ### Exam Strategy & Shortcut Use a quick visual chain calculation. Start from $1$, multiply by the last divisor, add the last remainder. Repeat moving upwards. $((1 \times 8 + 7) \times 5 + 4) \times 3 + 1 = 238$. Once you have $238$, rapidly divide it downwards by the reversed sequence. ### Common Pitfall The most common error is confusing "successive division" with standard division. Students frequently divide the original number directly by $8$, then directly by $5$, then directly by $3$, instead of dividing the *resulting quotients*. **Therefore, the correct answer is 6, 4, 2.**