Math 3200 Exam #2 Practice Problems
1. Suppose x ∈ R is positive. Prove that if x is irrational, then x1/6 is also irrational. Show that this is
not an if and only if statement by giving a counterexample to the converse.
2. Show that for any sets A and B,
(A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B).
3. Prove that, for any n ∈ N,
20 + 21 + 22 + . . . + 2n = 2n+1 − 1.
4. Suppose n ≥ 2 is an integer. Prove that there exists a ∈ N with 1 < a < n so that a2 ≡ 1 (mod n).
(Hint: What is a when n = 4? n = 5? What’s the pattern? )
5. Prove that 3 | (52n − 1) for all integers n ≥ 0.
6. Prove that the equation
2x3 + 6x + 1 = 0
has no integer solutions.
7. Prove that the equation from the previous problem does have a real solution.
8. Prove that there is no integer a so that a ≡ 2 (mod 6) and a ≡ 7 (mod 9).
9. Suppose x, y ∈ Z. Prove that if x2 (y 2 − 2y) is odd, then x and y are both odd.
10. Prove that for any positive x ∈ R,
x+
1
1
≥ 2.
x