History: On October 18, 1640, Fermat wrote a letter to Bernhard Frenicle de Bessy (1605–
1675), an official at the French mint who was a gifted student of number theory.
In his letter, Fermat communicated the following result(given as Theorem) Fermat did not provide a proof of this result but enclosed a note
promising that he would send along a proof, provided it was not too long. This theorem is known as Fermat's theorem.
Theorem: Let p be a prime and a any integer such that p and a are coprimes.
Then ap−1 ≡ 1 (mod p).
Information:
The first proof of Fermat's little theorem was given by Euler almost a century after Fermat's announcement. Leibniz had given same proof for Fermat's theorem almost 50 years prior to Euler but he didn't receive his share of credit.
=>This theorem can be used for questions like:
Q: Find the remainder when 241936 is divided by 17.
Ans: Here as 24 ≡ 7 (mod 17)
Therefore 241936 ≡ 71936 (mod 17)
But by Fermat's little theorem, 716 ≡ 1 (mod 17).
So,=7168121
71936 =716*121
≡ 1121 ≡ 1 (mod 17)
Thus, remainder is 1.
1675), an official at the French mint who was a gifted student of number theory.
In his letter, Fermat communicated the following result(given as Theorem) Fermat did not provide a proof of this result but enclosed a note
promising that he would send along a proof, provided it was not too long. This theorem is known as Fermat's theorem.
Theorem: Let p be a prime and a any integer such that p and a are coprimes.
Then ap−1 ≡ 1 (mod p).
Information:
The first proof of Fermat's little theorem was given by Euler almost a century after Fermat's announcement. Leibniz had given same proof for Fermat's theorem almost 50 years prior to Euler but he didn't receive his share of credit.
=>This theorem can be used for questions like:
Q: Find the remainder when 241936 is divided by 17.
Ans: Here as 24 ≡ 7 (mod 17)
Therefore 241936 ≡ 71936 (mod 17)
But by Fermat's little theorem, 716 ≡ 1 (mod 17).
So,=7168121
71936 =716*121
≡ 1121 ≡ 1 (mod 17)
Thus, remainder is 1.
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