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Unformatted text preview: MATH 332 - ALGEBRA AND NUMBER THEORY FIRST MIDTERM - PRACTICE TEST Note: (1) Calculators are not allowed in the exam. (2) You may assume the following axioms and theorems: (a) Axiom : The natural numbers N satisfies the Well Ordering Principle, i.e. every non-empty subset of natural numbers contains a least element. (b) 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 . Theory Problem 1. Prove that if p is prime and p | ab then either p | a or p | b . Explain why the previous statement can be re-written as follows: if p is a prime and ab 0 mod p then a 0 mod p or b 0 mod p (or equivalently, if ab in Z /p Z then either a or b in Z /p Z ). Theory Problem 2. Prove the Fundamental Theorem of Arithmetic, i.e. every natural number n > 1 is a product of primes, and the representation is unique, except for the order of factors....
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