A boy multiplies 987 by a certain number and obtains 559981 as his answer. If in the answer both 9's are wrong but the other digits are correct, then the correct answer will be

### Concept & Logic When a multiplication result contains errors in specific digits, we can use the fundamental **rules of divisibility** to identify the correct answer among the choices. The true product must be perfectly divisible by its original factors. ### Step-by-Step Solution * **Given:** Multiplier: $987$ Flawed Answer: $559981$ (The $5$, $5$, $8$, and $1$ are correct; the $9$'s are wrong). Target format: $55 \_ \_ 81$ * **Calculation / Deduction:** Analyze the known factor, $987$, for divisibility. The sum of the digits of $987$ is $9 + 8 + 7 = 24$. Since $24$ is divisible by $3$, the number $987$ is a multiple of $3$. Consequently, the final correct product must ALSO be a multiple of $3$. For a number to be divisible by $3$, the sum of its digits must be a multiple of $3$. Test the sum of the digits for the given options: * (a) $553681 \implies 5 + 5 + 3 + 6 + 8 + 1 = 28$ (Not divisible by $3$) * (b) $555181 \implies 5 + 5 + 5 + 1 + 8 + 1 = 25$ (Not divisible by $3$) * (c) $555681 \implies 5 + 5 + 5 + 6 + 8 + 1 = 30$ (**Divisible by $3$**) * (d) $556581 \implies 5 + 5 + 6 + 5 + 8 + 1 = 30$ (**Divisible by $3$**) We have narrowed it down to (c) and (d). Now, check which of these is actually a multiple of $987$ by doing a quick rough division or checking the difference. Difference between (d) and (c) is $556581 - 555681 = 900$. Since $900$ is not a multiple of $987$, they cannot both be multiples of $987$. Let's test $555681$ (Option c) by dividing it by $987$: $$ 555681 \div 987 = 563 $$ It divides perfectly with no remainder. ### Exam Strategy & Shortcut **Digital Sum / Divisibility Rule of 3:** Relying entirely on divisibility by $3$ eliminates two options immediately. Between the final two, rather than doing long division, you can check divisibility by $7$ (since $987$ is $3 \times 7 \times 47$). Option (c) $555681 \div 7 \implies 79383$ (exact). Option (d) $556581 \div 7 \implies$ leaves a remainder. Therefore, (c) is the only mathematically possible answer. ### Common Pitfall A time-consuming mistake is assuming you have to set up algebraic equations with variables for the missing digits (e.g., $550081 + 1000x + 100y = 987k$) and solving for them. Always leverage the provided multiple-choice options in conjunction with basic divisibility rules. ### Final Answer **Therefore, the correct answer is 555681.**