189-346B/377B: Number Theory

Practice Midterm Exam

Monday, February 5, 2006

1. Using only basic properties of the gcd, (in particular, without using the fundamental theorem of arithmetic), show that, if an integer n divides the product ab and gcd(a, n) = 1, then n divides b.

2. Find all the (positive and negative) integer solutions to the equation y^{2} − 2y = x^{3} in which y is odd.

3. Compute the value of the 11-th Fermat number 2^{2^{11}} + 1 modulo 17. You should express your answer as an integer between 0 and 16.

4. Let φ be Euler’s φ-function. Compute φ(108). Show that there is no primitive root modulo 108.

5. Show that the polynomial x^{2} + 1 has a root in Q_{5} and Q_{13} but not in Q_{7}.