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I want to find out what is$$2007^{2008}\pmod{1000}$$.

I used this website to find that the answer is $801$, but I'm not sure how they got there.

My attempt:

$2007^{2008}\pmod{1000}\equiv7^{2008}\pmod{1000}\equiv2401^{502}\pmod{1000}\equiv401^{502}\pmod{1000}\equiv160801^{251}\pmod{1000}\equiv801^{251}\pmod{1000}$

I pretty much gave up after this because it was getting too tedious and I didn't feel like this was the right approach. I got the $801$ but it has a power of $251$ which I don't know how to get rid of. Is there a quicker way to do this? If so, how? Thanks in advance!

Aiden Chow
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  • This post's How do I compute $a^b,\bmod c$ by hand? answers has quite a few good suggestions, of which you might find at least some of them helpful not only for this particular problem, as there's already $2$ answers for it specifically, but for other such similar problems you may encounter in the future. – John Omielan Feb 26 '20 at 05:40
  • If $7^4\equiv 201\pmod{1000}$ then $7^{4k} \equiv (200 + 1)^k \equiv \sum_{j=0}^k C_j* 200^j \pmod {1000}$. But if $j > 1$ then $200^j \equiv 0 \pmod{1000}$. So $(200+1)^k \equiv 200k + 1$ so $7^{20}\equiv (7^4)^5 \equiv (200+1)^5 \equiv 1\pmod {1000}$. So $2007^{2008}\equiv 7^8\equiv 201^2 \equiv 801$. Thats it. .... but read up on Euler's Theorem. – fleablood Feb 26 '20 at 06:45
  • @fleablood: $7^4\equiv\color{red}401\bmod1000$ – J. W. Tanner Feb 26 '20 at 10:59

2 Answers2

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By Euler's Theorem, we know that: $$7^{\phi(1000)}=7^{400}\equiv1\pmod{1000}$$ Therefore, $$2007^{2008}\equiv7^{2008}\equiv7^8\equiv801\pmod{1000}$$

MafPrivate
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Hint:

$$7^{2008}=(50-1)^{1004}\equiv(1-50)^{1004}\pmod{1000}$$

$$\equiv1-\binom{1004}150+\binom{1004}250^2\pmod{10^3}$$

Now $\displaystyle\binom{1004}2\equiv0\pmod2$

$\implies\displaystyle\binom{1004}250^2\equiv0\pmod{2\cdot50^2}\equiv0\pmod{1000}$

Alternatively, $7^4=(1+2400)$

$$7^{4n}=(1+2400)^n\equiv1+2400n\pmod{1000}$$

So, ord$_{1000}7\equiv5\cdot4,2008\equiv8\pmod{20}$

$$\implies7^{2008}\equiv7^8\pmod{1000}\equiv(1+2400)^2\equiv1+2\cdot2400+2400^2\equiv1+4800$$

lab bhattacharjee
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