Number Theory
# Basic Applications 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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