When a certain number is multiplied by 13, the product consists entirely of fives. Find the smallest such number.

### Concept & Strategy The problem implies an equation where $x \times 13 = 55555\dots$ To find the smallest integer $x$, we must perform long division of a sequence of 5s ($55555\dots$) by 13 until we reach a point where the remainder is exactly zero. ### Step-by-Step Solution - **Calculation / Deduction:** Set up a long division dividing a continuous string of 5s by 13. - Divide 55 by 13: Quotient is 4, Remainder is $55 - 52 = 3$. - Bring down the next 5 (making 35). Divide 35 by 13: Quotient is 2, Remainder is $35 - 26 = 9$. - Bring down the next 5 (making 95). Divide 95 by 13: Quotient is 7, Remainder is $95 - 91 = 4$. - Bring down the next 5 (making 45). Divide 45 by 13: Quotient is 3, Remainder is $45 - 39 = 6$. - Bring down the next 5 (making 65). Divide 65 by 13: Quotient is 5, Remainder is $65 - 65 = 0$. - Since the remainder is now 0, the division is complete. The quotient sequence forms our answer: 42735. ### Exam Strategy & Shortcut Perform standard division but watch the unit digits. Because the final division must leave a 0 remainder and the dividend brings down a 5, the last step of the division must involve a multiple of 13 that ends in 5. Since $13 \times 5 = 65$, you know the final digit of your answer will be 5 once the remainder hits 6. ### Common Pitfall Miscalculating a single subtraction during the long division will completely alter the subsequent remainders, leading to a never-ending cycle or a completely wrong quotient. Double-check your basic multiplication table for 13. ### Final Answer Therefore, the correct answer is **42735**.