UPSC CDS 2 2023 Math Paper

Concept:We need to find a 3-digit number divisible by 10 using the two given statements.Explanation:Let the 3-digit number be $100x + 10y + 0$, where $x$ is the hundred's digit and $y$ is the ten's digit. The unit digit is 0 because the number is divisible by 10. From Statement I: Interchanging hundred's and ten's digits gives the number $100y + 10x + 0$. The difference is $180$, so $(100x + 10y) - (100y + 10x) = 180$. This simplifies to $90x - 90y = 180$, or $x - y = 2$. Statement I alone gives only one equation, so it is not sufficient. From Statement II: Halve the hundred's digit and interchange the ten's and unit digits. The new number is $100(x/2) + 10 \cdot 0 + y = 50x + y$. The difference is $336$, so $(100x + 10y) - (50x + y) = 336$. This simplifies to $50x + 9y = 336$. Statement II alone also gives only one equation, so it is not sufficient. Using both statements together, we solve the two linear equations: $x - y = 2$ and $50x + 9y = 336$. Substitute $x = y + 2$ into the second equation: $50(y + 2) + 9y = 336$, giving $59y + 100 = 336$, so $59y = 236$, $y = 4$. Then $x = 6$. The number is $640$, which satisfies divisibility by 10. Thus, both statements together are necessary to answer the question.Answer:The question can be answered by using both the statements together.