UPSC CDS 1 2023 Math Paper

Concept:Divisibility by 25 requires the last two digits to be divisible by 25.Divisibility by 9 requires the sum of digits to be divisible by 9.Explanation:The number $277XY5$ is divisible by 25.Last two digits form $Y5$; the only possible values are $Y = 2$ or $Y = 7$.Statement I: The number is divisible by 9.Sum of digits = $2+7+7+X+Y+5 = 21 + X + Y$.If $Y=2$, sum = $23+X$; for divisibility by 9, $X=4$.If $Y=7$, sum = $28+X$; for divisibility by 9, $X=8$.Thus Statement I alone gives two possible $X$ values ($4$ and $8$), so it is not sufficient.Statement II: $X > 5$.Without any other condition, $X$ can be $6,7,8,9$ (since $X$ is a digit).Many numbers of the form $277X25$ and $277X75$ are divisible by 25 regardless of $X$.Hence Statement II alone is not sufficient.Combining both statements:From Statement I, $X = 4$ or $X = 8$.From Statement II, $X > 5$, which eliminates $X=4$.Therefore, $X = 8$ is uniquely determined.Answer:The question can be answered by using both statements together, but not by either statement alone. Option C.