Math 115 - Fall 2000 - Ribet - Final

# Math 115 - Fall 2000 - Ribet - Final - Let p be a prime...

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Math 115 Professor K. A. Ribet Final Exam December 13, 2000 This is a closed-book exam: no notes, books or calculators are allowed. Explain your answers as clearly and as completely as you possibly can. The blue book that you hand in at the end of the exam is your only representative when the exam is graded. 1 (4 points) . Find an integer x such that x 7 mod 37 and x 2 12 mod 37 2 . 2 (5 points) . Let n be an odd positive integer. Show that n is a perfect square if and only if b n = 1 for all integers b prime to n . 3 (5 points) . Express the infinite continued fraction [ 1 , 2 , 3] in the form a + b c with a , b and c integers. 4 (6 points) . Find a positive integer f so that x 271 f x mod 29 · 31 for all x prime to 29 · 31. 5 (4 points) . Decide whether or not 263 is a square mod 331. (Both numbers are primes.) 6 (5 points) . Use the equations 3469 = 2 · 1298 + 873 1298 = 1 · 873 + 425 873 = 2 · 425 + 23 425 = 18 · 23 + 11 to write 3469 / 1298 as a simple continued fraction. 7 (6 points)
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Unformatted text preview: . Let p be a prime that is congruent to 1 mod 4. View the quadratic residues (i.e., squares) mod p as integers between 0 and p-1. Show that the sum of these integers is p ( p-1 4 ). [Example: when p is 5, the residues are 1 and 4. Their sum is 5.] 8 (4 points) . If eggs in a basket are taken out 2, 3, 4, 5 and 6 at a time, there are 1, 2, 3, 4 and 5 eggs left over, respectively. If they are taken out 7 at a time, there are no eggs left over. What is the least number of eggs that can be in the basket? 9 (5 points) . Show that a positive integer n is a perfect number if and only if X d | n 1 d = 2. If n is a perfect number, show that tn is not a perfect number when t > 1. 10 (6 points) . Let n be a positive integer. Suppose that the Fermat number p = 2 2 n + 1 is prime. Prove that 3 ( p-1) / 2 ≡ -1 mod p ....
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