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What is the units digit of 3 to the 2004th power?

 Apr 6, 2020
 #1
avatar+1088 
+7

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.

 Apr 6, 2020
 #2
avatar+24979 
+2

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

 

laugh

 Apr 7, 2020
edited by heureka  Apr 7, 2020

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