The number of digits in the smallest number, which when multiplied by 7 yields all nines, is

### Concept & Strategy Similar to the previous problem, finding a number that yields a sequence of repeating digits when multiplied by a prime requires dividing the repeating sequence by that prime until there is no remainder. We are looking for $N$ such that $N \times 7 = 99999...$ Therefore, $N = \frac{99999...}{7}$ ### Step-by-Step Solution * **Given:** Target product sequence is $9999...$ Divisor is $7$. * **Calculation / Deduction:** Divide a sequence of $9$'s by $7$ step-by-step until the remainder hits zero. * $9 \div 7 = 1$ (Remainder $2$) * Bring down $9 \implies 29 \div 7 = 4$ (Remainder $1$) * Bring down $9 \implies 19 \div 7 = 2$ (Remainder $5$) * Bring down $9 \implies 59 \div 7 = 8$ (Remainder $3$) * Bring down $9 \implies 39 \div 7 = 5$ (Remainder $4$) * Bring down $9 \implies 49 \div 7 = 7$ (Remainder $0$) The division terminates here. The smallest number $N$ is $142857$. The quotient $142857$ is composed of $6$ digits. ### Exam Strategy & Shortcut **Cyclic Number Memorization:** A powerful shortcut for bank and SSC exams is memorizing the reciprocal of $7$. $$ \frac{1}{7} = 0.142857142857... $$ This repeating decimal block has exactly **6 digits**. Because it repeats every $6$ digits, multiplying $142857$ by $7$ will yield $999999$ (a number with six $9$'s). You can instantly answer "6" without doing any division. ### Common Pitfall A major reading comprehension pitfall is finding the smallest number ($142857$) and then looking for it in the options, getting confused when it isn't there. The question explicitly asks for the **"number of digits"** in that number, not the number itself. ### Final Answer **Therefore, the correct answer is 6.**