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

elementary-number-theory modular-arithmetic exponentiation decimal-expansion

Solution 1:

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)

Solution 2:

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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