On multiplying a number by 7, all the digits in the product appear as 3's. The smallest such number is

### Concept & Strategy This is a divisibility problem disguised as a multiplication problem. If an unknown number multiplied by $7$ yields a string of $3$'s ($33333...$), then we can find the number by simply dividing $33333...$ by $7$ until we reach a remainder of $0$. ### Step-by-Step Solution * **Given:** $x \times 7 = 33333...$ We need to find the smallest integer $x$. * **Calculation / Deduction:** Perform standard long division of $33333...$ by $7$: * $33 \div 7 = 4$ with a remainder of $5$. (First digit of quotient is **$4$**) * Bring down the next $3$, making it $53$. * $53 \div 7 = 7$ with a remainder of $4$. (Next digit is **$7$**) * Bring down the next $3$, making it $43$. * $43 \div 7 = 6$ with a remainder of $1$. (Next digit is **$6$**) * Bring down the next $3$, making it $13$. * $13 \div 7 = 1$ with a remainder of $6$. (Next digit is **$1$**) * Bring down the next $3$, making it $63$. * $63 \div 7 = 9$ with a remainder of $0$. (Next digit is **$9$**) Since the remainder is now $0$, the division is complete. The quotient is $47619$. ### Exam Strategy & Shortcut **Unit Digit and Option Verification:** You can solve this by looking at the options and working backwards. The target number ends in a $3$. All options end in a $9$. ($9 \times 7 = 63$, which gives the correct unit digit $3$). Instead of long division, pick an option and multiply it by $7$ mentally or on paper. Let's test option (a) $47619$: $$ 47619 \times 7 = 333333 $$ It perfectly matches the condition. ### Common Pitfall Students often get intimidated by the "infinite" looking sequence of 3's and attempt complex algebraic formulations. Simply setting up a continuous division is the intended and fastest mechanical path. ### Final Answer **Therefore, the correct answer is 47619.**