Fermat's little theorem gives a condition that a prime
must satisfy:
Theorem. If P is a prime, then for any integer A,
( A^{P}  A ) must be divisible by P.
Let's check:
2^{9}  2 = 510, is not divisible by 9, so it
cannot be prime.
3^{5}  3 = 240, is divisible by 5, because 5 is
prime.
Presentation Suggestions:
This may be a good time to explain the difference
between a necessary and sufficient condition.
The Math Behind the Fact:
This theorem can be used as a way to test if a number is
not prime,
although it cannot tell you if a number is prime.
Fermat's theorem is a special case of a result known as Euler's theorem:
that for any positive integer N, and any integer A relatively prime to N:
( A^{phi(N)}  A ) must be divisible by N,
where phi(N) is Euler's totient function that returns the number of positive integers less than or equal to N that are relatively prime to N. So when N is prime, phi(N)=N.
Fermat's "little" theorem should not be confused with Fermat's Last Theorem.
How to Cite this Page:
Su, Francis E., et al. "Fermat's Little Theorem."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
