On 7/5/2022 10:40 AM,
henh...@gmail.com wrote:
(7 ^ 2022) --- What are the least 3 digits?
( (7 ^ 4 ) ^ 2022)) --- What are the least 3 digits?
(7 ^ ( 4 ^ 2022)) --- What are the least 3 digits?
(or ... What is it [ mod 1000 ] ?)
[spoiler space]
Using a spreadsheet to compute (7 ^ n) mod 1000 for n = 1, 2, ...., we
find that 7 ^ 20 = 1 (mod 1000). Thus:
7 ^ 2022 = 7 ^ 2 (mod 1000), so it ends in 049.
(7 ^ 4) ^ 2022 = 7 ^ 8088 = 7 ^ 8 (mod 1000), so it ends in 801.
For the last one, compute (4 ^ n) md 20 for n = 1, 2, ..., and note that
4 ^ 3 = 4 (mod 20). Thus:
7 ^ (4 ^ 2022) = 7 ^ (4 ^ 2) (mod 1000), so it ends in 601.
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