UPSC CDS I 2024 Math Paper

Concept:The Highest Common Factor (HCF) is the product of the common prime factors taken with their smallest exponents present in all numbers.Explanation:Prime factorisation of 768: $768 = 2^{8} \times 3$.Given HCF: $32xy = 2^{5} \times x \times y$.The other number is $x^{6}y^{2}$.Since the HCF includes $x$ and $y$ as factors, $x$ and $y$ must be prime numbers that are common to both numbers. The only prime factors in 768 are 2 and 3.Hence $x$ and $y$ must be 2 and 3 in some order.For the exponent of 2: 768 has $2^{8}$, and the HCF has $2^{5}$. In $x^{6}y^{2}$, the exponent of 2 comes from $x^{6}$. If $x=2$, then $2^{6}$ gives at least $2^{5}$; if $y=2$, the exponent from $y^{2}$ is $2^{2}$, which is not enough to match $2^{5}$ from the HCF. So $x=2$ and $y$ must be 3 to supply the factor 3 (since 768 has $3^{1}$).Check: $x=2, y=3$ gives $x^{6}y^{2}=2^{6}\cdot 3^{2}$, HCF is $2^{5}\cdot 3=96$? Wait, recalc: HCF(768, 576) = $2^{6}\cdot 3$? Actually common powers: $2^{5}$ from 32xy? Let's compute properly: 768 = $2^{8}\cdot 3$, $x^{6}y^{2}=2^{6}\cdot 3^{2}$. Minimum exponents: $2^{6}$ and $3^{1}$ gives $64\cdot 3=192$. But $32xy = 32\cdot 2\cdot 3=192$. Yes correct.Therefore $x=2$, $y=3$.Sum $x+y = 2+3 = 5$.Answer:$5$