UPSC CDS I 2024 Math Paper

Concept:The $100^{\text{th}}$ place is the third digit from the right (the hundreds place). For numbers ending with $25$, any power with exponent $\ge 2$ always ends with the last three digits $625$.Explanation:Observe the pattern for powers of $25$: $25^{2}=625$, $25^{3}=15625$, $25^{4}=390625$.In each case, the last three digits are $625$.Now, $225$ ends with $25$. So $(225)^{2}=225^{2}=50625$ (ends with $625$).For $(225)^{n}$ where $n \ge 2$, the last three digits remain $625$ because the multiplication only involves factors that preserve the $625$ ending.Thus, for $(225)^{40}$, the last three digits are $625$.The $100^{\text{th}}$ place corresponds to the hundreds digit, which is $6$.Answer:$6$