# How to find the last digit of $3^{1000}$?

I'm thinking of modulo, but really don't know how to start? A hint would be sufficient.

Thanks,
Chan

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$3^4 \equiv 1 \bmod 10$ – user17762 Feb 26 '11 at 23:43
@Sivaram Ambikasaran: Thanks ;) – Chan Feb 26 '11 at 23:44

HINT $\rm\ \ mod\:\ 10\::\ \ 3^2 \equiv -1\ \Rightarrow\ 3^4 \equiv 1\$ so you need only consider the exponent $\rm\ (mod\ 4)$

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Thanks, totally forgot that a number is in base 10. – Chan Feb 26 '11 at 23:44

Lets look at the last digits of the first few powers:

The last digit of $3^0$ is $1$

The last digit of $3$ is $3$

The last digit of $3^2$ is $9$

The last digit of $3^3$ is $7$

The last digit of $3^4$ is $1$

The last digit of $3^5$ is $3$

The last digit of $3^6$ is $9$

Notice a pattern? Why does this pattern exist? What is going on when I multiply by three? Based on this we could guess that it has a period of $4$ so that $3^{4n}\equiv 1$.
Use this to find the last digit of $3^{1000}$.

(Do you know modular arithmetic? If so it is a lot easier)

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I do these types of problems$\mod 4$

So, I look at $2000\mod 4$

$2000\mod 4\equiv 0 \mod 4$

I then do $3^0$, which ends in a $\boxed{1}$

QED!

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But why does this work? (Some thing is hidden here. Making it explicit might help.) – user21436 Mar 29 '12 at 19:04