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Fermat little theorem

Fermat little theorem

Fermat's little theorem was coined by Pierre de Fermat in 1640. Fermat little theorem states that

If p is a prime number, then for any integer a, the number ap - a is an integer multiple of p.

In modular arithmetic it is expressed as

ap = a (mod p)

For example, a = 3, p = 5, 35 = 243, 243 - 3 = 240 = 5 x 48 is the integer multiple of 5.

If a is not divisible by p, Fermat's little theorem is equivalent to the statement that ap-1 - 1 is an integer multiple of p:

ap-1 = a (mod p)

For example, a = 3, p = 5, 35-1 = 81 = 81 – 1 = 80 = 5 x 16

Euler given the proof for this theorem in 1736. Fermat theorem become one of the important fundamental result of Elementary number theory. Fermat little theorem is different from the Fermat last theorem (also called as Fermat conjecture)

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