If the exercise has parts (a), (b), (c), etc., label each part of your solution and include the exercise questions for these parts as well. Provide any formulas you'll use in your computation(s) or summarize your strategy in a sentence prior to any computations. This will help the reader know why you're performing the computation. Show all steps using material from this class. If you use material from another class that isn't a prerequisite, provide the relevant definition or theorem so that someone from this class could learn from your solution. Think about this step as showing me how much of the material you've learned, or write a solution so that another student in class who doesn't know how to solve the exercise could learn from your solution. Answers without justification will receive 0 points. Write a conclusion either using a sentence (preferred for an application exercise, like a word problem) or symbolically to summarize what your work shows.

2. Let f : Z Z, f(n) is equal to the number of divisors of n. Note that d is a divisor of m if nee: (a) Give the domain, codomain, and range of f. (b) Compute f(10) and f(20). Then give a formula for f(2n) and f(4n) where n N. (c) Is f injective? Justify. (d) Is f surjective? Justify. (e) Is f invertible? Justify. (f) A prime element of Z is an integer p {1,0,1} and such that if p is a divisor of a product of integers a- b, then p is a divisor of a or b. Give three examples of primes Pi, p2,p3 in Z and compute f (pi), f(p2), and f(p3). What do you notice? If p is a prime number in Z, can you find a formula for f(p)? Explain