Euler's Theorem

Fermat's Little Theorem

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

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

12^6 \bmod 7 = 1

14^7 \bmod 8 = 1

16^8 \bmod 9 = 1

18^9 \bmod {10} = 1

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$\begin{aligned} 1^4 \bmod 5 &=& 1 \\ 2^4 \bmod 5 &=& 1 \\ 3^4 \bmod 5 &=& 1 \\ 4^4 \bmod 5 &=& 1 \end{aligned}$

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

$5^4 \bmod 5 = 1$

Yes, it is true No, it is not true

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True or false?

$42^6 \bmod 7 = 1.$

True False

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What is the remainder when $3^{456}$ is divided by 7?

1 2 3 4

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$\large 32^{23} \pmod {23} =\, ?$

5 7 9 11

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