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Difference between revisions of "Mock AIME 4 2006-2007 Problems/Problem 7"
(→‎Solution)
 
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</cmath>
 
</cmath>
  
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Now, <math>3^{87}=(3^{20})^4\cdot 3^7\equiv 401^4\cdot 187\pmod{1000}\equiv 601\cdot 187\pmod{1000}\equiv \boxed{387}\pmod{1000}</math>.
+
Now,
 +
 
 +
<cmath>\begin{align*}3^{87}=(3^{20})^4\cdot 3^7&\equiv 401^4\cdot 187\pmod{1000} \\
 +
&\equiv 601\cdot 187\pmod{1000} \\
 +
&\equiv \boxed{387}\pmod{1000}.
 +
\end{align*}</cmath>
 +
 
 
== See also==
 
== See also==
  

Latest revision as of 23:15, 4 January 2010

Problem

Find the remainder when $3^{3^{3^3}}$ is divided by 1000.

Solution

Using the Carmichael function, we have $\lambda(1000)=100$, so $3^{100}=1\pmod{1000}$. Therefore, letting $N=3^{3^3}$, we seek to find an $n$ such that $N\equiv n\pmod{100}$ so that $3^N\equiv 3^n\pmod{1000}$.

Using the Carmichael function again, we have $\lambda(100)=20$, so $N=3^{27}\equiv 3^7\pmod{100}\equiv 87\pmod{100}$. Therefore $n=87$, and so we have the following: \[3^{3^{3^3}}\equiv 3^{87}\pmod{1000}.\]

Now,

\begin{align*}3^{87}=(3^{20})^4\cdot 3^7&\equiv 401^4\cdot 187\pmod{1000} \\ &\equiv 601\cdot 187\pmod{1000} \\ &\equiv \boxed{387}\pmod{1000}. \end{align*}

See also

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