First Prelim

# First Prelim - MATH 332 ALGEBRA AND NUMBER THEORY FIRST...

• Test Prep
• 2

This preview shows pages 1–2. Sign up to view the full content.

MATH 332 - ALGEBRA AND NUMBER THEORY - FIRST MIDTERM - 10/02/2007 Show all work. No calculators. There are 2 theory questions and 4 additional problems. You may assume the following axioms and theorems: (1) Axiom : The natural numbers N satisfy the Well Ordering Principle, i.e. every non- empty subset of natural numbers contains a least element. (2) Theorem: Let a, b, c be integers. The linear equation ax + by = c has a solution if and only if gcd( a, b ) divides c . (3) Theorem: Let p be a prime and let a, b be any integers. If p | ab then p | a or p | b . More generally, if p | a 1 a 2 · · · a k then p divides some a i . Just in case you need them, the following are all the primes below 100: 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 . ———————– Theory Question 1. (20 points) Prove the uniqueness part of the Fundamental Theo- rem of Arithmetic , i.e. assume that every natural number n has a factorization and then prove that this factorization is unique up to the order of the factors.

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern