Find the smallest number which when multiplied by 9 gives the product as 1 followed by a certain number of 7s only.

### Concept & Logic We are looking for a multiplier that produces a number of the format $1777\dots7$, which is a multiple of 9. The divisibility rule for 9 dictates that the sum of the digits of a number must be a multiple of 9 for the entire number to be divisible by 9. ### Step-by-Step Solution - **Calculation / Deduction:** Let the number of 7s be $n$. The sum of the digits is $1 + 7n$. - Test values for $n$ to find the smallest sum that is a multiple of 9: - $n = 1$: Sum = $1 + 7 = 8$ - $n = 2$: Sum = $1 + 14 = 15$ - $n = 3$: Sum = $1 + 21 = 22$ - $n = 4$: Sum = $1 + 28 = 29$ - $n = 5$: Sum = $1 + 35 = 36$ - 36 is divisible by 9. Therefore, the smallest valid product requires five 7s, making the number 177777. - To find the multiplier (the original number), divide the product by 9: $$ 177777 \div 9 = 19753 $$ ### Exam Strategy & Shortcut Alternatively, use option elimination. Multiply the given options by 9 and check the last few digits. $19753 \times 9$: $3 \times 9 = 27$ (ends in 7). $5 \times 9 + 2 = 47$ (ends in 7). This confirms the sequence of 7s without having to find the large target number first. ### Common Pitfall A frequent error is successfully identifying the target number (177777) but selecting it as the answer (if it were an option). The question asks for the *smallest number which when multiplied*, meaning you must find the multiplier by dividing by 9. ### Final Answer Therefore, the correct answer is **19753**.