Three boys A, B, C were asked to divide a certain number by $1001$ by the method of factors. They took the factors in the orders $13$, $11$, $7$; $7$, $11$, $13$ and $11$, $7$, $13$ respectively. If the first boy obtained $3$, $2$, $1$ as successive remainders, then find the successive remainders obtained by the other two boys B and C.

### Concept & Strategy The "method of factors" relies on successive division. Just like standard successive division, you reconstruct the original dividend by working backward from the final quotient using the first boy's divisors and remainders. Then, apply the specific divisor sequences for the other boys to find their respective remainders. ### Step-by-Step Solution **Calculation / Deduction:** * **Step 1: Reconstruct the original number using Boy A's data.** * Boy A divisors: $13$, $11$, $7$. * Boy A remainders: $3$, $2$, $1$. * Assume the final quotient is $1$. Work backward: $$ z = (7 \times 1) + 1 = 8 $$ $$ y = (11 \times 8) + 2 = 90 $$ $$ x = (13 \times 90) + 3 = 1173 $$ * The original number is $1173$. * **Step 2: Calculate Boy B's successive remainders.** * Boy B divisors: $7$, $11$, $13$. * $1173 \div 7$: Quotient = $167$, Remainder = $4$. * $167 \div 11$: Quotient = $15$, Remainder = $2$. * $15 \div 13$: Quotient = $1$, Remainder = $2$. * Boy B's remainders: $4, 2, 2$. * **Step 3: Calculate Boy C's successive remainders.** * Boy C divisors: $11$, $7$, $13$. * $1173 \div 11$: Quotient = $106$, Remainder = $7$. * $106 \div 7$: Quotient = $15$, Remainder = $1$. * $15 \div 13$: Quotient = $1$, Remainder = $2$. * Boy C's remainders: $7, 1, 2$. ### Exam Strategy & Shortcut To quickly reconstruct the number, use the bottom-up sequence calculation: $(((1 \times 7) + 1) \times 11 + 2) \times 13 + 3 = 1173$. Once established, treat the rest of the problem as two quick successive division drills. Keeping your scratchpad organized with a simple bracket format prevents cascading arithmetic mistakes. ### Common Pitfall A critical error occurs if a student interprets "method of factors" as dividing the original number by $7$, $11$, and $13$ independently rather than sequentially updating the dividend to the new quotient. This yields completely incorrect remainders. **Therefore, the correct answer is B: 4, 2, 2; C: 7, 1, 2.**