=========================preview======================
(math110)[2007](f)final~PPSpider^_10426.pdf
Back to MATH110 Login to download
======================================================
HKUST
MATH110 CONCEPTS IN MATHEMATICS
Final Examination Name:
20th December 2007 Student I.D.:
16:30C18:30

Directions:
.
Do NOT open the exam until instructed to do so.

.
Please write your name, ID number, and Section in the space provided above.

.
Answer ALL questions.

.
This is a closed book examination.

.
You may write on both sides of the examination papers.

.
You must show the working steps of your answers in order to receive full marks.

.
All mobile phones and pagers should be switched o. during the examination.

.
Cheating is a serious o.ense. Students who commit this o.ense may receive zero mark in the examination. However, more serious penalty may be imposed.


Question No. Points Out of
Q. 1 15
Q. 2 18
Q. 3 10
Q. 4 19
Q. 5 18
Q. 6 15
Total Points 95

. To verify your identity, you must sign your name here:
Answer each of the following questions.

1. (a) Prove by considering an integer of the form N = p! + 1 that there exist in.nitely many primes. [8]
Ans.

(b) Let ak (k =0, 1, m) be integers such that 0 ak < 10 and let S =(amam.1 a1a0)10,
T = a0 . a1 + a2 . +(.1)m am.
Prove that S and T always have the same remainder when divided by 11. [7]

2. (a) Let a, b be two given integers. Write down precisely the de.nition of their greatest common divisor d = gcd(a, b) [2]
Ans.

(b) Show that if a and b are given integers, not both zero, then the set
H = ax + by : x, y are integers
is precisely the set of all integral multiples of d = gcd(a, b). [6]

2. (cont.)

(c) Find integers x and y so that [5]

gcd(1769, 2378) = 1769x + 2378y.

Ans.

(d) Let a, b, c be three given integers. We de.ne their greatest common divisor D = gcd(a, b, c) as we have done for a, b in (a). Show that there exist integers x, y, z such that D = ax + by + cz. Please give a full justi.cation of your steps. [5] 3. (a) State a criterion of when exactly we can .nd all solutions of the equation ax b (mod n) where a, b, n are integers. [2]
Ans.

(b) Find all solutions to the equation 660x 595 (mod 1385). [8]

4. (a) Let Bn = [1/n, 1 . 1/n2] for each n 2. What is ? Justify your answer. [7]
n=2Bn
Ans.

(b) De.ne what is the least upper bound for a given set of real numbers. [2]

Ans.

(c) Let A = {0} where the is de.ned in part (a). Does the set A has a
n=2 Bnn=2Bn
greatest lower bound and least upper bound, minimum and maximum, and if so what are they? [5]
Ans.

(d) Prove your assertion for the least upper bound of A in part (c). [5]

5. Prove the existence of the real number 321 . [18]
Ans.

Question 5 (cont.):
6. (a) Let w0,w1, ,wn and z0,z1, ,zn be two given sequences of numbers, and that p(x) is
a polynomial of degree n that satis.es f(zj)= wj,j =0, 1, ,n. Prove that
n
. w(x)
p(x)= wk .
(x . zk)w.(zk)
k=0
where w(x)=(x . z0)(