UPSC CDS I 2019 Math Paper

Concept:The expression $5^{2n} - 2^{3n}$ can be rewritten as $(5^2)^n - (2^3)^n = 25^n - 8^n$.For any positive integer $n$, $a^n - b^n$ always has a factor $(a - b)$.Explanation:Rewrite the given expression: $5^{2n} - 2^{3n} = (5^2)^n - (2^3)^n = 25^n - 8^n$.Using the algebraic identity $a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + \dots + b^{n-1})$, we see that $(a - b)$ is always a factor.Here $a = 25$ and $b = 8$.Therefore, $(25 - 8) = 17$ must be a factor of $25^n - 8^n$ for any positive integer $n$.Check with a simple value: put $n = 1$, then $5^{2} - 2^{3} = 25 - 8 = 17$, which is divisible by 17.Thus, 17 is a common factor for all $n$.Answer:17 (Option C)