+1072 Units Digit of the Powers of 3:
3^1=3
3^2=9
3^3=7
3^4=1
3^5=3
Now we see that the units digit of 3 repeats every 4 powers. 4 goes into 2004 without any remainder. The units digit of 3^4 is 1, therefore the units digit of 3^2004 is 1.
+26393 What is the units digit of 3 to the 2004th power?
\(\begin{array}{|rcll|} \hline && \mathbf{3^{2004} \pmod{10}} \\ &\equiv & 3^{2*1002} \pmod{10} \\ &\equiv & \left(\mathbf{3^2}\right)^{1002} \pmod{10} \quad & | \quad \mathbf{3^2 \equiv -1 \pmod{10}} \\ &\equiv & \left(-1\right)^{1002} \pmod{10} \\ &\equiv & \mathbf{1 \pmod{10} } \\ \hline \end{array}\)
The units digit of 3 to the 2004th power is 1
