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# Number Theory

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What is 3^{-1}+3^{-2} mod 25? Express your answer as an integer from 0 to 24, inclusive.

I know that 3^{-1}=17, but what does 3^{-2} mean?

I had an account but lost it :(

Jul 30, 2019
edited by Guest  Jul 30, 2019

#1
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um nevermind..........

Jul 30, 2019
#3
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OK, young person, here is my interpretation of your question:

3^(-1) mod 25 is the "modular multiplivative inverse of 3 mod 25".

3^(-2) =1 / 9, which can be written as 9^(-1) mod 25. So then, you have:

[3^(-1) + 9^(-1)] = 12^(-1) mod 25 = 23 - which is the modular multiplicative inverse of 12 mod 25.

Jul 30, 2019
#4
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What is $$3^{-1}+3^{-2} \pmod {25}$$

$$\begin{array}{|rcll|} \hline && \mathbf{3^{-1}+3^{-2} \pmod {25}} \\\\ &\equiv& 3^{-1}+ \left(3^{-1}\right)^2 \pmod {25} \\ \hline \end{array}$$

Modular multiplivative inverse $$\mathbf{3^{-1}\pmod {25}}$$ :

$$\begin{array}{|rcll|} \hline && \mathbf{3^{-1}\pmod {25}} \\ &\equiv& 3^{(\varphi{(25)}-1)} \pmod {25} \quad | \quad \varphi{(25)} = 20\quad \varphi(25) = 25*\left(1-\dfrac{1}{5} \right) \\ &\equiv& 3^{(20-1)} \pmod {25} \\ &\equiv& 3^{19} \pmod {25} \\ &\equiv& 1162261467 \pmod {25} \\ &\equiv&\mathbf{ 17 \pmod {25}} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline && \mathbf{3^{-1}+3^{-2} \pmod {25}} \\\\ &\equiv& 3^{-1}+ \left(3^{-1}\right)^2 \pmod {25} \\ &\equiv& 17+ 17^2 \pmod {25} \\ &\equiv& 306 \pmod {25} \\ &\equiv& \mathbf{6 \pmod {25}} \\ \hline \end{array}$$ Jul 31, 2019
edited by heureka  Jul 31, 2019