The last four digits

Find the last four digits of 10 1 999 101^{999} .


The answer is 9901.

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2 solutions

Mathh Mathh
Oct 10, 2015

By Binomial Theorem: ( 100 + 1 ) 999 ( 999 1 ) 100 1 998 + 1 999 99901 9901 ( m o d 10000 ) (100+1)^{999}\equiv \binom{999}{1}\cdot 100\cdot 1^{998}+1^{999}\equiv 99901\equiv 9901\pmod{10000} .

Dillon Chew
Nov 23, 2015

See pattern. 101 x 101 last 4 digits are 0201. 101 x 101 x 101 last 4 digits are 0301... thus 101^999 would be 9901 :)

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