Find the remainder when 2^1990 is divided by 1990.

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Find the remainder when 2^1990 is divided by 1990.

Mathematics
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Mod arithmetic :') @terenzreignz
Why me? :/

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Other answers:

Because you.
In this question How to square to so much power ?
Might have to resort to totients..... @ParthKohli ?
Sir if i use the exponent rule will it work here ?
Ah! Euler's Theorem!
factor 1990 first
Time to doodle... \[\large 2^{1990}=4^{995}\]
How to factor it ?
how to factor 1990?
What is 4^5? \[\Large = 1024^{199}\]
try \(2\times 5\times 199\)
1990 = 10*199 = 2* 5* 199
\[\large 1024^{199}=1024\cdot 1024^{198}=1024\cdot 2048^{99}\]
try \(2\times 5\times 199\) means ?
Now let's start working some "mod magic" and reduce the bases at mod 1990 \[\Large =_{(mod \ 1990)} \ \ 1024\cdot 58^{99}\]
|dw:1367281405845:dw|
ok So LCM can be take as a good way of factorizing numbers ok ?
\[\Large \equiv 1024\cdot 58 \cdot 58^{98}\]
it is the best way!!!
\[\Large \equiv 1682 \cdot 1374^{49}\] gah... possibly inefficient...
ok
\[\Large\equiv 678\cdot (-616)^{48}\]
This is daunting.
\(1990=2\times 5\times 199\) and \(2^2\equiv 4(5) \) also \(2^{198}\equiv 1(199)\) by fermat
Mods Sir(s) I have to make you know that the Equation you are posting have not opened yet
actually this is kind of a pain isn't it
if you are seeing "math processing error", try refreshing
What to do ?
\[\Large \equiv 678\cdot (1356)^{24}\]
Yo It's done...Mods you'll great REFRESHING WORKED. No SEE
Now I can see
\[\Large \equiv 678 \cdot (-634)^{24}\equiv678\cdot (634)^{24}\]
2^1990 is divided by 1990 what to actually for this ?
2^1990 2^11 = 2048 = 38 mod 1990 2^(11(180)+10) is what i had in mond :)
I lack creativity, guys :)
@mikaela19900630 you may too se the question I have posted now.
Thank you all of you. I can do these type of question Now :)
*from Now
\[\Large \equiv 678\cdot (-24)^{24}\equiv 678\cdot 24^{24}\]
cr*p... sorry \[\Large \equiv 678\cdot (-24)^{\color{red}{12}}\equiv 678\cdot 24^{\color{red}{12}}\]
Thank You Very Much.

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