189-235A: Algebra 1 Practice Midterm Exam

1. Recall that the power set of a set A, denoted P (A), is the set of all subsets of A. Let f : A → P (A) be a function from A to its power set. Show that the subset B of A defined by

B = {a ∈ A such that a ∉ f(a)}

is not an element of the image of f.

2. Compute the greatest common divisor of the integers

a = 13200000008150000000000000000132,
b = 1320000000815.

3. Compute the least residue modulo N = 95 of the integer

3^{11000000000000000000000000000000000000000000}

4. Using only the basic properties of the gcd proved in class, (and not the fundamental theorem of arithmetic) show that if a p is a prime and p divides a product ab of two integers, then p necessarily divides either a or b.