Office hours and practice final

I will be available for office hours from 10-1 on Tuesday and Wednesday. HOWEVER these are not drop-in times -- if you want to see me, email me (ideally the night before, or at least half an hour in advance).

I will be putting up practice final questions tonight, with answers posted tomorrow night. Here are the first few:

1a. Let a,b be integers such that a² = b³.
Is b necessarily a square?
If yes, prove it, otherwise, give a counterexample (a pair {a,b}).

1b. Let a,b be integers such that a¹⁰ = b¹⁵.
Is b necessarily a 10th power?
If yes, prove it, otherwise, give a counterexample (a pair {a,b}).

1c. Let a,b be integers such that a³²³ = b³⁹¹.
Then a,b are both powers of some other integer. What are those powers?

2. Prove by induction that sum_i=1...n 1/i(i+1) = 1 - 1/(n+1).

3. Let k,n be natural numbers, such that n! > kⁿ. Show that for all N > n, N! > k^N.

4. Let R,S be two equivalence relations on the same set X.

a. Show that R intersect S is an equivalence relation (where "intersect" means as sets of ordered pairs).

b. Let RuS denote the union of R and S. Of the three conditions required of an equivalence relation, which ones does RuS automatically satisfy? For each condition {reflexive, symmetric, transitive}, either show that RuS satisfies it, or give an example of X,R,S where RuS doesn't satisfy it.

5. "Let b,c,n be integers, such that bⁿ > cⁿ. Then for all N>n, b^N > c^N." State the negation of this statement and prove that.