UPSC CDS 2 2022 Math Paper

Concept:The LCM of two numbers is given. Find all possible pairs $(x,y)$ with $x>y$ whose LCM is $481$. Then determine which statement uniquely identifies the pair, allowing calculation of $(3x-2y)$.Explanation:Factor $481$ into its prime factors: $481 = 13 \times 37$ or $481 = 481 \times 1$.Since LCM is $481$ and $x>y$, the only possible pairs are $(x,y) = (481,1)$ and $(37,13)$.Now test each statement:Statement 1: $y>1$. This eliminates $(481,1)$ because $y=1$ is not $>1$. Only $(37,13)$ remains. Then $(3x-2y) = 3(37)-2(13)=111-26=85$. So Statement 1 alone gives a unique value.Statement 2: HCF of $x$ and $y$ is $1$. Both pairs have HCF $1$: $\text{HCF}(481,1)=1$ and $\text{HCF}(37,13)=1$. Two different values for $(3x-2y)$ exist: for $(481,1)$ it is $1441$, for $(37,13)$ it is $85$. Hence Statement 2 alone is insufficient.Therefore, only Statement 1 alone is sufficient to answer the question.Answer:Option A: Statement-1 alone is sufficient to answer the question.