Wednesday, 29 July 2015

number theory - Fermat's Little Theorem definition clarification.



Fermat's Little Theorem states that (acc to Gallian book)




a^p \mod p= a \mod p.



Does it mean that we get the same remainder when both a^p and a are divided by some prime p? I am quite confused about this statement. Through wikipedia,
I read a^p \equiv a \mod p. Kindly help. I am new to this number system topic.


Answer



a^p\mod p\equiv a \mod p\implies a^p-a\equiv 0 \mod p \implies p\text{divides} (a^p-a)\implies a^p-a=kp\implies a^p=kp+a



So what is the remainder when a^p is divided by p


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