Euler Function

(See attached file for full problem description) --- We consider the special case when m=3 and n=4. (a) Write down the correspondence between numbers in and pairs of integers in given by the function f. In other words, write out the 12 values f(a) where . (b) Fore each value you computed above, circle the equations that correspond to . (c) How does this set of ordered pairs compare with ? Note: In this discussion will be the function defined in the chapter summary for the Chinese Remainder Theorem given by . Note: g is defined as follows: g: Where is the multiplicative inverse of m1 modulo m2 and conversely.

Euler Function

(See attached file for full problem description) --- We consider the special case when m=3 and n=5. (a) Find the explicit function from the Chinese Remainder Theorem Chapter summary. (Recall that g is the inverse function of f.) (b) Write down all ordered pairs (a,b) Є . (c) Compute g(a,b) for each ordered pair in part (b), reducing each answer to its remainder modulo 15. (d) Compare the list in part (c) with the integers modulo 15 which are relatively prime to 15. Note: In this discussion will be the function defined in the chapter summary for the Chinese Remainder Theorem given by . Note: g is defined as follows: g: Where is the mu... click for more

Primitive roots

(See attached file for full problem description with all symbols) Suppose d and n are integers greater than 1 such that . If a is an integer relatively prime to n, show that .

Primitive Roots

(See attached file for full problem description with all symbols) --- Suppose that n is odd and a is a primitive root modulo n. (a) Show that there exists and integer b such that and . (b) Show that b is a primitive root modulo 2n.

Primitive Roots

(See attached file for full problem description) --- Assume r and s are relatively prime positive integers and that n=rs. Let and assume that gcd(a,n)=1. Prove: (a) (b)