Find the roots: 3(7+15−3) -4(5−37+1) =11 x≠35,−1/7
Answers
Answer:
There's number theory theorem that states ap−1=1mod(p)ap−1=1mod(p) (Fermat's theorem).[Assuming p doesn't divide a] So 10p−1=1mod(p)10p−1=1mod(p). [This won't hold for p=2 and p=5 because 2 and 5 divide 10]. So if you divide 1 by p you will get a 1 in the p- 1 reiteration. So if this is the first one, the period is p-1. If this isn't the first 1 then the period divides p-1.
A more basic explanation is that when you divide p into 1 you get a remainder. When you divide p into that you get another. There are only p-1 possible remainders (other than 0) so eventually you have to repeat remainders (or come to a 0 and stop). When you repeat you cycle through so at most the cycle is p-1. That doesn't explain why if the cycle is less than p-1 then it divides p-1, however.