Homework 6

Due Date: Friday, March 11

1. Find the smallest positive integer \(x\) for which \[ x \;\equiv\; 7814845152\pmod{11830911289} \] and \[ x \;\equiv\; 903346502\pmod{18592481077} \]
2. Let \(p\) be an odd prime. Prove there there are infinitely many natural numbers \(n\) for which \(n2^n + 1\) is a multiple of \(p\).
3. Prove that if \(n>6\) then \(\phi(n) > \sqrt{n}\).