Practice Prelim 1 Solutions

Practice Prelim 1 Solutions - MATH 332 ALGEBRA AND NUMBER...

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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 0 in Z /p Z then either a 0 or b 0 in Z /p Z ). Proof. Suppose p divides ab but p does not divide a . Then gcd( p, a ) = 1 (otherwise, there is d > 1 such that d | p and d | a , and since p is prime d = p but p does not divide a ). By the theorem above, there exist x, y Z such that ax + py = 1 . Multiplying this equation by b gives: abx + pby = b. Since p divides ab and p obviously divides pb , then p divides any linear combination of ab and pb . Hence p divides b = ( ab ) x + ( pb ) y . The rest of the problem follows from the fact that p | a if and only if a 0 mod p . / 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. Proof. See the book or your class notes. / Theory Problem 3. Prove Euclid’s theorem on the infinitude of primes, i.e. prove that there exist infinitely many prime numbers. Proof. See the book or your class notes. / Theory Problem 4. Write precise statements for the following theorems (you do not need to prove them): (1) Wilson’s Theorem. (2) Fermat’s Little Theorem. Proof. See the book or your class notes. Remember that in class we said that Wilson’s theorem is an if-and-only-if statement. / Problem 1. Use Euclid’s algorithm to: (1) Find the greatest common divisor of 13 and 50 . (2) Find all solutions of the linear diophantine equation 13 x + 50 y = 2 . 1
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2 MATH 332 - ALGEBRA AND NUMBER THEORY FIRST MIDTERM - PRACTICE TEST (3) Find the multiplicative inverse of 13 modulo 50 . Find the multiplicative inverse of 50 modulo 13 . Can you use your previous work to find the multiplicative inverse of 7 modulo 27 ?
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