A number of two digits has 3 for its unit's digit, and the sum of digits is $\frac{1}{7}$ of the number itself. The number is

### Concept & Formula Algebraic representation of two-digit numbers. A two-digit number with ten's digit $t$ and unit's digit $u$ is always mathematically expressed as $10t + u$. ### Step-by-Step Solution Let the ten's digit of the number be $x$. We are given that the unit's digit is 3. So, the exact number is represented as: $10x + 3$. The sum of the digits is: $x + 3$. According to the problem, the sum of the digits is $\frac{1}{7}$ of the number itself: $x + 3 = \frac{1}{7}(10x + 3)$ Multiply both sides by 7 to eliminate the fraction: $7(x + 3) = 10x + 3$ $7x + 21 = 10x + 3$ Rearrange to solve for $x$: $21 - 3 = 10x - 7x$ $18 = 3x$ $x = 6$ The ten's digit is 6, and the unit's digit is 3. Therefore, the number is 63. ### Exam Strategy & Shortcut Direct option testing is dramatically faster here because all options already end in 3. You just need to check the core condition: $\text{Sum of digits} = \frac{\text{Number}}{7}$ (which means Number must be a multiple of 7). * (a) 43 is not divisible by 7. * (b) 53 is not divisible by 7. * (c) 63 is divisible by 7. Check the sum: $6 + 3 = 9$. And $63 \div 7 = 9$. It matches perfectly! * (d) 73 is not divisible by 7. ### Common Pitfall Setting up the algebraic equation incorrectly. Many students write the number simply as $x + 3$ instead of the correct place-value representation $10x + 3$, which completely breaks the subsequent algebra. ### Final Answer Therefore, the correct answer is **63**.