• #### ( (7 ^ 4 ) ^ 2022)) --- What are the least 3 digits? [ mod 1000 ]

From henhanna@gmail.com@21:1/5 to All on Tue Jul 5 10:40:22 2022
(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 ] ?)

( nice puzzle ... if i could 've posted it yesterday )

------- pls ( pls ^2 ) wait a few days before posting answers or hints.

--- SoupGate-Win32 v1.05
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• From Edward Murphy@21:1/5 to henh...@gmail.com on Fri Jul 8 06:58:27 2022
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.

--- SoupGate-Win32 v1.05
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