UPSC CDS 1 2022 Math Paper

Concept:Divisibility rules and properties of odd/even numbers help verify statements about cube sums and powers.Explanation:For statement 1: Let three consecutive natural numbers be $n$, $n+1$, and $n+2$.Sum of cubes: $n^3 + (n+1)^3 + (n+2)^3$.For $n=1$: $1 + 8 + 27 = 36$, which is divisible by $9$.For $n=2$: $8 + 27 + 64 = 99$, divisible by $9$.For $n=4$: $64 + 125 + 216 = 405$, divisible by $9$.General proof: Expand to get $3n^3 + 9n^2 + 15n + 9 = 3(n^3+3n^2+5n+3)$. Since each term is a multiple of $3$, and the sum is also a multiple of $9$ for all natural $n$, statement 1 is correct.For statement 2: An odd number $>1$ can be written as $2k+1$.Even power: $(2k+1)^{2m}$.Check small even powers: $3^2=9$, $5^2=25$, $7^2=49$.When divided by $8$, each gives remainder $1$: $9-8=1$, $25-24=1$, $49-48=1$.General: $(2k+1)^2 = 4k(k+1)+1$, and $k(k+1)$ is even, so $4k(k+1)$ is divisible by $8$. Hence any even power also leaves remainder $1$. So statement 2 is correct.Answer:Both statements 1 and 2 are correct.