Again I am back with my lame questions....they are so lame that you would doubt if I am a legit level 5 In Number theory.

Find 3^2012's last 2 digits and also try to find its last 3 digits is the question

I know its mod 100 but i am not able to simplify it.

I would love if someone could tell me NOT ONLY HOW TO SOLVE THIS, BUT ALSO HOW TO SOLVE ANY SUCH PROBLEM USING MOD 100, 1000....AND ALSO HOW TO DO MODULAR EXPONENTIATION FAST. Thanks

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## Comments

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TopNewestI have completely different approach to the question to make it look very simple, \({ 3 }^{ 2012 }\quad could\quad be\quad written\quad as:\\ { { (3 }^{ 4 }) }^{ 503 }\\ it\quad is\quad quite\quad interesting\quad to\quad note\quad that\quad 81\quad multiplied\quad by\quad 81\quad henceforth\quad changes\\ its\quad ten's\quad digit\quad after\quad 5\quad cyclic\quad repetitions\quad i.e.\quad 8,6,4,2,0,8........\\ unit\quad digit\quad will\quad always\quad be\quad 1\\ \Rrightarrow divide\quad power\quad by\quad 5\quad i.e.\frac { 503 }{ 5 } R=3\\ find\quad the\quad third's\quad ten's\quad digits\quad i.e.\quad 4\\ answer\quad will\quad be \boxed{41}\)

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If you're not already aware, @Finn Hulse created this Modular Arithmetic set for you.

For further resources, you can look at #Modular Arithmetic, or do a search and filter by tags. @Akshaj Kadaveru 's note on Modular Arithmetic is a detailed introduction.

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Akshaj is the bomb diggity. :D

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I somehow don't really understand that comment.

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Yes, sir. I am aware of this. But thanks to Nathan Ramesh I came to know of the Euler Theorem. Since then, i learnt a lot more about this topic and am yet learning. And I haven't seen a note about Euler's theorem anywhere, thus didnt know till recently. Thanks :)

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Check out my set Olympiad Number Theory for similar notes.

You could also do a search for Euler's Theorem, and filter by notes, which will display my note on Euler's Theorem.

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@Calvin Lin - I extremely apologize for the inadvertent error of mine while setting up the answer of the question- "Integral Divisors". Your answer was right. Kindly change it as required. Thanks

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@Finn Hulse @Daniel Liu @Nathan Ramesh please help me out..is it some kind of a cycle or is it modular exponentiaiton...please tell me! :)

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Use something called eulers totient theorem, which states \(3^{2012}\equiv 3^{2012\pmod{\phi(100)}}\equiv 3^{2012\pmod{40}}\equiv 3^{12}\equiv (3^3)^4\equiv 27^4\equiv 729^2\equiv 29^2\equiv 41 \pmod{100}\)

Where \(\phi(n)\) denotes the number of numbers less than \(n\) that are relatively prime to \(n\).

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Thanks :) ,....Thanks a lot!!!!!!!...Could you tell me how to learn more about these? For eg:- I knew this function but not this theorem :(....So any form of resource would be appreciated. @Nathan Ramesh

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@Nathan Ramesh - Take a look at this one- calculating \( 7^{1728} mod 1000\) . Using the euler theorem, you get that phi (1000)= 400. so this is equal to \( 7^{128} mod1000\) . How do I simplify it further.? @Finn Hulse you too have a look at this please. And i would like some further genralization on how to simplify such large mod powers. :)

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@Nathan Ramesh

OK. Why do the first 62 terms cancel...I didnt get it could you explain againLog in to reply

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@Nathan Ramesh

Okay, the last two don't and all I have to do is find their sum and get it mod 1000 ...rite?Log in to reply

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@Nathan Ramesh

Wow! Thank you. So they're like 64c63 into 50 and 1 into 1...so it becomes 3201 mod 1000 equal to 201. but answer is 801...am i wrong somewer?Log in to reply

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@Nathan Ramesh @Finn Hulse pL answer

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