What will be the last three digits of the number $17^{256}$?
This question really intrigued me. First I thought that binomial theorem would help. But miserably failed. The given answer is 681. Please help!
2026-03-30 20:44:13.1774903453
What will be the last three digits of the number $17^{256}$?
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It can be easily done by binomial $$17^{256}=(289)^{128}=(290-1)^{128}$$ In the expansion all terms except last three are multiple of 1000. Hence last three digit are by last three digits of last 3 terms i.e Last 3 digit of ${128 \choose 126} (290)^2-{128 \choose 127} (290)+{128 \choose 128}$ $$. $$ =Last three digits of above terms are $(800-120+1)$ $$=681$$