Fermat's Little Theorem states that if p is a prime number, then
for any integer a, the number is an
integer multiple of p. It can also be expressed as
.
For the deduction part one need to know what is d?
Discrete structures please help!! Use Fermat's little theorem to find the remainder when 91000 is divided by 13. To get credit, use Fermat's little theorem and show how each step is done without using a calculator.
7. Use Fermat's Little Theorem to find the remainders of each of the division problems a. 6150 -19 b. 937531.
Determine the tens digit of 3^3^100 . (Hint: Use Euler's generalization of Fermat's little theorem)
solve number #7 please.
domain is a field. 7. State Fermat's theorem and use it to find the remainder when 31233 is divided by 11.
Solve the following problems using Fermat's Little Theorem (a) Prove that, if 5 does not divide n, then 5n1. (b) Prove that, if gcd(n, 6) 1, then 12n2 - 1 (c) Prove that, if 5 does not divide n-1, , or n+1, then 5(n21).
r proof of Fermat's little theo- 2. Use Corollary 3.6 to give anothe Proposition 3.3 of Chapter 1. (Hint: In our more up-to-date language, the theorem should be restated as follows: given any prime number p, a. a for all a E Zp.) rem, Corollary 3.6. If IGI n, and a E G is arbitrary, then ane. Proof. Let the order of the element a be k. By Corollary 3.4, k n, so there is an integer e with n...
the second part of the question can be solved by the chineses
remainder theorem.
Problem 4 (4pts) Recalled Fermat's little theorem: For every p, a € N, if p is a prime and pla, then -I = 1 mod p. Use Fermat's little theorem to find a = 71002 mod 13) and b =(71002 mod 41). Find an 3 (0 < < 13 x 41) such that r = a (mod 13) and 2 = b (mod 41).
prove 10.8-10.9
LLLLLLLLL think the converse to Fermar's Little Theorem is true? 10.8 Theorem. Lern be a natural number greater than 1. Then 7 is prime if and only if a"- = 1 (mod n for all natural numbers a less than n. 10.9 Question. Does the previous theorem give a polynomial or exponen- rial time primalin test? Inventing polynomial time primality tests is quite a challenge. One way to salvage some good from Fermat's Little Theorem is to weaken...
7. EXTRA CREDIT [5 points] For Zp, where p is a prime, Fermat's theorem gives us an alternative way to compute the multiplicative inverse of any given nonzero [x]p: raise it to the power of p-1. Show that Fermat's theorem is a corollary (a special case) of Euler's theorem, i.e., show how one can derive the former from the latter.
please
complete exercises 10.4, 10.5, 10.6, 10.7 and 10.9, thank you so
much! (I dont understand your comment what is qs 3.6?)
10.4 Exercise. Show that the algorithm descrihed in Question 3.6 for com puting a (mod n) is a polynomial time algorithm in the number of digits in r In the next scrics of problems you will cxplore the usc of this opcration as a means of testing for primality by starting with a familiar theorem. Theorem (Fermat's Little...