Use Exercise 37 to show that the integers 235 - 1, 234 -

Chapter 4, Problem 38E

(choose chapter or problem)

Problem 38E

Use Exercise 37 to show that the integers 235 - 1, 234 - 1, 233 - 1, 231 - 1, 229 - 1, and 223 - 1 are pairwise relatively prime.

Reference Exercise 37:

        Use Exercise 36 to show that a and b are positive integers, then gcd(2a - 1, 2b - 1) = 2gcd(a,b) - 1. [Hint: Show that the remainders obtained when the Euclidean algorithm is used to compute gcd(2a - 1, 2b - 1) are from 2r - 1, where r  is a remainder arising when the Euclidean algorithm is used to find gcd(a,b).]

Reference Exercise 36:

        Show that if a and b are both positive integers then, (2a - 1) mod (2b - 1) = 2a mod b -1.