Just 3 digits
Find the last 3 digits in the decimal representation of:
1 2 6 1 6 2 2 1 6 2 6 1 6 2 1 6 1 2
The answer is 376.
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2 solutions
We are looking for 1 2 6 1 6 2 2 1 6 2 6 1 6 2 1 6 1 2 m o d 1 0 0 0 . As 1 0 0 0 = 1 2 5 ⋅ 8 and 1 2 6 1 6 2 2 1 6 2 6 1 6 2 1 6 1 2 = 1 m o d 1 2 5 and 1 2 6 1 6 2 2 1 6 2 6 1 6 2 1 6 1 2 = ( − 2 ) 1 6 2 2 1 6 2 6 1 6 2 1 6 1 2 = 0 m o d 8 , by the Chinese Remainder Theorem, there exists a unique solution smaller than 1000. By quickly testing 1 , 1 2 5 + 1 , 2 5 0 + 1 , 3 7 5 + 1 , . . . , we find that the solution is 3 7 6 .
Last 3 digits of 1 2 6 3 are 3 7 6 and 1 2 6 × 3 7 6 has last 3 digits 3 7 6 hence 1 2 6 n has last 3 digits 3 7 6 for n ≥ 3