Fermat's little theorem
Proof
If \(a\) is an integer, \(p\) is a prime number, and \(a\) is not divisible by \(p,\) then \(a^{p - 1} \equiv 1 \pmod{p}\)
If \(a\) is an integer, \(p\) is a prime number, and \(a\) is not divisible by \(p,\) then \(a^p \equiv a \pmod{p}\)
Using the theorem
Find the remainder:
\(2^{50} \div 17\)
\(4\)
Find the remainder:
\(4^{532} \div 11\)
\(5\)
Find the remainder:
\(7^{2001} \div 5\)
\(2\)
Find the remainder:
\(5^{3571} \div 11\)
\(5\)
Find the remainder:
\(4^{100{,}000} \div 19\)
\(4\)