Number Theory
Euler's 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?
\[ \large 32^{23} \pmod {23} =\, ?\]