Math 115 - Fall 2000 - Ribet - Final

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

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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)
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern