Assignment # 4-Number Theory

(1) Use Fermat’s Little Theorem to compute 31^100 (mod 19).
(2) For which values of n is 3^n + 2^n divisible by 13?
(3) Show that no prime number of the form 4k + 3 can divide a number of the form
n^2 + 1.
(4) Find all primes p such that p | 2p + 1.
(5) Prove that there are inffinitely many positive integers n such that phi(n) is a perfect
square.
(6) Use Euler’s Theorem to compute 7^8^9 modulo 100.
(7) Bonus: Prove that for every even positive integer n, n^2 – 1 divides 2^n! – 1.

assignment 4 solution

assign4soln