Assignment 6 (pdf)

1. Using the fact that \(4\times 15 = 60\), solve the equation \(15x + 12 \equiv 3\bmod 59\). Your solution should be a number \(x\) computed "modulo 59". It should be an integer between 0 and 58.
2. Find the first power of \(3\) that is congruent to 1 modulo 11, and use this information to find out the value of \(3^{2014}\bmod 11\).
3. True or False: For every \(y\) we can solve the equation \(5x\equiv y \bmod 23\) for \(x\).
4. True or False: For every prime \(p > 3\) the equation \(x^2 + 1\equiv 0\bmod p\) has a solution.
5. Find an \(x\) such that \(x = 10\bmod 13\) and \(x = 5\bmod 59\).