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Number Theory

Euler's Theorem

Fermat's Little Theorem

         

Fermat's little theorem states that if aa and pp are coprime positive integers, with pp prime, then ap1modp=1a^{p-1} \bmod p = 1 .

Which of the following congruences satisfies the conditions of this theorem?

14mod5=124mod5=134mod5=144mod5=1 \begin{aligned} 1^4 \bmod 5 &=& 1 \\ 2^4 \bmod 5 &=& 1 \\ 3^4 \bmod 5 &=& 1 \\ 4^4 \bmod 5 &=& 1 \end{aligned}

We are given that the 4 congruences above are true. Is the following congruence true as well?

54mod5=1 5^4 \bmod 5 = 1

True or false?

426mod7=1.42^6 \bmod 7 = 1.

What is the remainder when 34563^{456} is divided by 7?

3223(mod23)=? \large 32^{23} \pmod {23} =\, ?

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