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## Basic Applications of Modular Arithmetic

Solve integer equations, determine remainders of powers, and much more with the power of Modular Arithmetic.

# 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?

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

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

$5^4 \bmod 5 = 1$

True or false?

$42^6 \bmod 7 = 1.$

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

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

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