Advanced Mathematics

1. Let a, b, c be integers. Suppose that a divides bc and that (a, b) = 1. Prove that a divides c

X 3. Let n = pt¹ × p²² × × per, where r, all ei, 1 ≤ i ≤r, are natural numbers, and all pi, 1 ≤ i ≤r are distinct primes. How many ordered pairs (u, v) of positive integers are there with the property that lcm(u, v) = n, where lcm stands for least common multiple.

8. How many commutative binary relations on a set of cardinality n N are there? (A binary relation on a set A is a any mapping A × A → A. By custom, rather than (a, b) € A, we usually write a o b; a binary relation "o" is called commutative if a ob= boa for all a, b = A.)

6. In how many ways can five indistinguishable rooks be placed on an 8-by-8 chessboard so that no rook can attack another and neither the first row nor the first column is empty?

4. Suppose we have 35 indistinguishable things arranged in order along a straight line and suppose you select some 9 of them. (a) In how many ways can this be done? (b) This time when selecting them you need a gap of at least one thing between any two consecutive things selected. How many ways are there for such a selection? (c) What if it is at least two gaps?

5. Generalize all three parts of the previous problem to an arbitrary number of things given, an arbitrary number of things selected, with gaps of arbitrary length.

Q 11 Use the graph to find: (a) The numbers, if any, at which f has a local maximum. What are these local maxima? (b) The numbers, if any, at which f has a local minimum. What are these local minima?

Starting with the graph of a basic function, graph the following function using the techniques of shifting. compressing, stretching, and/or reflecting. Find the domain and range of the function. g(x)=(x-1)³-5

5. Prove that every integer of the form 6n - 1 for n E N has at least one prime factor congruent to 5 mod

2. Where on the path r(t) = (t²-5t)i + (21+1)] +38² k vectors orthogonal? are the velocity and acceleration