UPSC CDS 1 2022 Math Paper

Concept:Express both terms as powers of the same base to factor and identify a clear factor.Explanation:Rewrite $27^{5}$ as $(3^{3})^{5} = 3^{15}$.The expression becomes $3^{15} + 3^{13}$.Factor out the lower power $3^{13}$: $3^{13}(3^{15-13} + 1) = 3^{13}(3^{2} + 1)$.Simplify $3^{2} = 9$, giving $3^{13}(9 + 1) = 3^{13} \times 10$.Since $3^{13} \times 10$ is a multiple of $10$, it is divisible by $10$.None of the other options ($8$, $12$, $21$) are factors: $8$ requires $2^{3}$, $12$ requires $2^{2} \times 3$, and $21$ requires $3 \times 7$.The number $3^{13} \times 10$ has only one factor of $2$ (from $10$) and no factor of $7$, so only $10$ divides it.Answer:10