UPSC CDS 1 2022 Math Paper

Given: $p$ and $q$ are the LCM and $\text{HCF}$ of two positive numbers $p: q=14: 1, p \cdot q=1134$ Formula Used: $\text{LCM} \times \text{HCF}=$ Product of numbers Calculation: Let the value of $p$ and $q$ be $14 x$ and $x$ respectively According to the question $pq=1134$ $\Rightarrow (14 x)(x)=1134$ $\Rightarrow 14 x^{2}=1134$ $\Rightarrow x^{2}=81$ $\Rightarrow x=9$ So, $p=14 \times 9=126$ And $q=1 \times 9=9$ Since $p$ and $q$ are the $\text{LCM} \& \text{HCF}$ of two numbers Then We can let two numbers are $9 m$ and $9 n$ [ since 9 is the HCF of two numbers] Now, According to the formula $\text{LCM} \times \text{HCF}=$ Product of numbers $\Rightarrow 126 \times 9 = 9 m \times 9 n$ $\Rightarrow mn=14$ So, The possible value of $m$ and $n$ is $(14,1) \& (7,2)$ Case I: When we take $m=14$ and $n=1$, we get Numbers are $9 \times 14=126$ and $9 \times 1=9$ So, Difference between the number is $126-9=117$ But we won't take this as it is not available in the options. Case II: If we take $m=7$ and $n=2$, we get Numbers are $9 \times 7=63$ and $9 \times 2=18$ So, Difference between the number is $63-18=45$ $\therefore$ The required difference is 45 .