CTET Paper II (Mathematics and Science) 4 Jan 2022 Solved Paper

Concept:LCM takes the highest power of each prime factor. HCF takes the lowest common power.Explanation:Given: $x = 2^{5} \times 3^{2} \times 5^{3}$, $y = 2^{4} \times 3^{3} \times 5^{2}$, $z = 2^{3} \times 3^{3} \times 5^{2}$.HCF: For each prime, pick the smallest exponent common to all three numbers.$2$: smallest is $3$ (from $z$). $3$: smallest common is $2$ (from $x$). $5$: smallest common is $2$ (from $y$ and $z$).So HCF $= 2^{3} \times 3^{2} \times 5^{2}$.LCM: For each prime, pick the largest exponent among the three numbers.$2$: largest is $5$ (from $x$). $3$: largest is $3$ (from $y$ and $z$). $5$: largest is $3$ (from $x$).So LCM $= 2^{5} \times 3^{3} \times 5^{3}$.Now divide LCM by HCF: $\frac{2^{5} \times 3^{3} \times 5^{3}}{2^{3} \times 3^{2} \times 5^{2}} = 2^{2} \times 3^{1} \times 5^{1} = 4 \times 3 \times 5 = 60$.Answer:C. 60