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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 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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