Prove that Iog2 3 is irrational leg stands for logarithm wit

Prove that Iog_2 3 is irrational (leg, stands for logarithm with base 2.) Prove that for every positive integer n 1 middot 2 middot 3 + 2 middot 3 middot 4 + ... + n(n + 1) (n + 2) = n(n + 1)(n + 2)(n + 3)/4 Show that the set S defined by 1 S and s + t S whenever s S and t S is the set of positive integers. Show that for any integer n > 13, there exist nonnegative integers x and y such that n = 4x + 5y. Use the modular exponentiation algorithm to find 3^2016 mod 11. Find s and t such that s middot 252 + t middot 356 = gcd(252, 356). (Start with computing gcd(252, 356) using the Euclidean algorithm.) Find the inverse of 24 modulo 31. Fun problem: show that if a and b are both positive integers, then (2^a - 1) mod 1) = (2^b - 1) = 2^a mod b - 1.

Solution

Solving first 2 questions as this is a multiple problem question

12) Using Euclid\'s method,

GCD = 4

Hence s = -24 and t = 17

13)

1 = (-9)*24 + 7*31 mod 31

Hence the inverse is -9

356 = 252*1 + 104

252 = 104*2 + 44

104 = 44*2 + 16

44 = 16*2 + 12

16 = 12*1 +4

12 = 4*3 + 0