TU Wien:Discrete Mathematics VO (Gittenberger)/Written Exam 2022-09-30

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1. Use the Chinese remainder theorem to solve the following system of congruence relations: 3x = 12 mod 13, 5x = 7 mod 22, 2x = 3 mod 7

2. Compute the number of integers x with 1 <= x <= 1000000 where x is neither a square nor a 3rd, 4th, or 5th power of some positive integer y.

3. List all elements of Z_3[x]/(x^2 + x + 1). Examine whether or not this is a field and whether or not x + 1 is a unit in it. If x + 1 is a unit, compute its multiplicative inverse.

4. List all sets S such that ({1,2,3}, S) is a matroid.