Fermat’s Little Theorem

Fermat’s little theorem gives a condition that a prime must satisfy:

Theorem. If P is a prime, then for any integer A, (AP – A) must be divisible by P.

Let’s check: 
29 – 2 = 510, is not divisible by 9, so it cannot be prime. 
35 – 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: (Aphi(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. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Lesley Ward

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