Other 320's Midterm from this term

Mathematics 320 Midterm Examination, Spring 2006 [Marchisotto]
Each question has a value of 40 points. Please show all your work.
1. For each of the following circle your answer and give a justification:
(a) Consider the linear Diophantine equation 10x + 14y = 232. How many
solutions exist?
infinitely many
(b) Let p1, p2, ..., pr be primes. Then p1|(p1p2...pr + 1).
(c) Let a, b, c  Z. If a| bc, then a| b or a|c. True
(d) Let n be an integer that is not divisible by 5. Then n2 – 1 is not
divisible by 5. True False.
2. Let R be a relation from N to N defined in the following way: a R b iff {p: p is
a prime and p|a} = {p: p is a prime and p|b}.
(a) Show R is an equivalence relation on Z.
(b) Find x, y  [12]R
3. Prove the following statement for natural numbers using the principle of
mathematical induction:
½ + 2/22 + 3/23 + … + n/2n = 2 – (n + 2)/2n
4. Let f and g be functions defined on R 
f = {(x, 3x – 1)} and g = {(x, x2 – 3}.
a. Prove f is one-to-one and onto.
b. Prove that g is neither one-to-one nor onto.
5. Use the definition for limit of a sequence to prove that the sequence
<an> for which an = n/(n2 + 1) has the limit 0.