I'm thinking of modulo, but really don't know how to start? A hint would be sufficient.
Thanks,
Chan
I'm thinking of modulo, but really don't know how to start? A hint would be sufficient.
Thanks,
Chan
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!
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)
HINT $\rm\ \ mod\:\ 10\::\ \ 3^2 \equiv -1\ \Rightarrow\ 3^4 \equiv 1\ $ so you need only consider the exponent $\rm\ (mod\ 4)$