novaBBS - rec.puzzles - ( (7 ^ 4 ) ^ 2022)) --- What are the least 3 digits? [ mod 1000 ]

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

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Subject: ( (7 ^ 4 ) ^ 2022)) --- What are the least 3 digits? [ mod 1000 ]
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by: henh...@gmail.com - Tue, 5 Jul 2022 17:40 UTC

(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.

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

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From: emurph...@zoho.com (Edward Murphy)
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Subject: Re: ( (7 ^ 4 ) ^ 2022)) --- What are the least 3 digits? [ mod 1000 ]
Date: Fri, 8 Jul 2022 06:58:27 -0700
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by: Edward Murphy - Fri, 8 Jul 2022 13:58 UTC

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.

backups: always in season, never out of style.

interests / rec.puzzles / ( (7 ^ 4 ) ^ 2022)) --- What are the least 3 digits? [ mod 1000 ]

Subject	Author
( (7 ^ 4 ) ^ 2022)) --- What are the least 3 digits? [ mod 1000 ]	henh...@gmail.com
Re: ( (7 ^ 4 ) ^ 2022)) --- What are the least 3 digits? [ mod 1000 ]	Edward Murphy