Mock AIME 4 2006-2007 Problems/Problem 7

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,

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

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Mock AIME 4 2006-2007 Problems/Problem 7

Problem

Solution

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