At least one of these is a Nice puzzle.
2 ^ 100 --- What is it mod 2016 ?
2 ^ 2022 --- What are the least 2 digits?
(or ... What is it mod 100 ?)
2 ^ 100 --- What is it mod 2016 ?
2 ^ 2022 --- What are the least 2 digits?
"henh...@gmail.com" <henh...@gmail.com> writes:
At least one of these is a Nice puzzle.Different shortcuts make them very easy to do manually.
2 ^ 100 --- What is it mod 2016 ?2016 = 2*1008 = 4*504 = 8*252 = 16*126 = 32*63
so we only need to calculate 2^100 mod 63, the mod 32 bit's easy
2^6=64 == 1, so 2^96 == 1
2^100 == 2^4 * 2^96 == 2^4 == 16 (mod 63)
16 != 0 mod 32 so add a multiple of 63 to correct that
63 == -1 (mod 32), so we want 16+16*63 = 16*64 = 1024
2^100 == 1024 (mod 2016)
2 ^ 2022 --- What are the least 2 digits?powers of 2 mod 25 go:
(or ... What is it mod 100 ?)
1, 2, 4, 8, 16, 7, 14, 3, 6, 12, -1, ... it's a primitive root, period 20 2022 = 101*20+2 so we want 2^2 == 4 (mod 25)
conveniently, 4 == 0 mod 4
so 2^2022 == 4 mod (100)
GP verification once the manual graft's done:
? Mod(2,2016)^100
Mod(1024, 2016)
? Mod(2,100)^2022
Mod(4, 100)
Phil
2 ^ 100 --- What is it mod 2016 ?
2 ^ 2022 --- What are the least 2 digits?
(or ... What is it mod 100 ?)
Sysop: | Keyop |
---|---|
Location: | Huddersfield, West Yorkshire, UK |
Users: | 322 |
Nodes: | 16 (0 / 16) |
Uptime: | 148:13:51 |
Calls: | 7,072 |
Calls today: | 1 |
Files: | 12,493 |
Messages: | 5,499,283 |