UPSC CDS 1 2022 Math Paper

Concept:
Natural number: All the positive integers from 1 to infinity.
Even number: Whole numbers that have the digits 0,2,4,6, or 8 in their one's place are even numbers.
Odd number: Any number which cannot be divided by two.
Calculation:
Statement: 1
Let n,n+1, and n+2 be three consecutive natural numbers.
Let, P(n)=n3+(n+1)3+(n+2)3 is divisible by 9 .
For n=1,
⇒13+23+33
⇒1+8+27=36‌ is divisible by ‌9
⇒‌ For ‌n=4
⇒43+53+63
⇒64+125+215=405 is divisible by 9
Hence, statement 1 is correct.
Statement: 2
Take 32,52,72 we get
⇒9,25,49 which is divisible by 8 and gives 1 as remainder.
Similarly, we can check for any other value as well.
Hence, statement 2 is correct.
∴ Both 1 and 2 are correct.