$$700^{1734} \mod{347}$$
I know how this could be calculated if I had $376$ in the exponent, using Fermat's theorem. But I have no idea how to approach this problem. What theorem's are appropriate to refer to?
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$700 \equiv 6 \mod 347$, so $700^{1734}\equiv 6^{1734} \mod 347$ – Michael Biro Nov 15 '14 at 19:13
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1Note that $1734=(346)(5)+4$. – André Nicolas Nov 15 '14 at 19:16
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See How do I compute $a^b,\bmod c$ by hand? – Bart Michels Apr 14 '15 at 08:29
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$347$ is prime, so you can use Fermat's little theorem. $$347 \nmid a \Longrightarrow a^{347-1} \equiv 1 \pmod{347}$$
Note that $700^{1734} = 700^{1730}\cdot 700^{4} = \left(700^{346}\right)^{5} \cdot 700^{4}$. And from Fermat's theorem $$\left(700^{346}\right)^{5} \cdot 700^{4} \equiv (1)^5 \cdot 700^{4} \pmod{347}$$
And now obvious steps $700^4 \equiv 6^4 \pmod{347} \wedge 6^4 \equiv 255 \pmod{347}$, so number you are looking for is 255.
However, if $347$ will be just coprime with $700$ (assume $347$ isn't prime) than you can use Euler's theorem. Of course this theorem works here too (example). In fact Fermat's little theorem is just a special case of Euler's totient theorem, but using whole is a little bit triumph of form over content.
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Hint : $347$ is a prime.
Also, $700\equiv 6\pmod{347}$
By eulers $6^{\phi(347)}\equiv 1\pmod{347}$ where $\phi$ is the Euler's phi function.
Swapnil Tripathi
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