Math 115 - Fall 2000 - Ribet - Midterm 1

# Math 115 - Fall 2000 - Ribet - Midterm 1 - 2 Find the...

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

Math 115 Professor K. A. Ribet First Midterm Exam September 29, 2000 This is a closed-book exam: no notes, books or calculators are allowed. Explain your answers in complete English sentences. No credit will be given for a “correct answer” that is not explained fully. In general, there is no need to simplify numerical answers. 1 (5 points) . Let a and b be positive integers for which a 4 divides b 3 . Prove that a divides b . 2 (10 points) . Let f ( x ) = x 2 - x - 1. Here are some values of f : i 0 1 2 3 4 5 6 7 8 9 10 ··· f ( i ) - 1 - 1 1 5 11 19 29 41 55 71 89 ··· . Find integers a and b so that f ( a ) and f ( b ) are both divisible by 11 2 but so that a - b is not divisible by 11
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2 . Find the number of solutions mod 5 ยท 11 2 to the congruence f ( x ) โก 0 mod 5 ยท 11 2 . 3 (3 points) . Let m = 173 ยท 193. Find positive integers a and b with โ m < b < m + 1 2 for which m = b 2-a 2 . 4 (5 points) . Use the identity 1 = 89 ยท 24-61 ยท 35 ( * ) to solve the simultaneous congruences x โก n 3 mod 89 12 mod 61. 5 (4 points) . Using (*), ๏ฌnd integers a and b with 1 = 24 a +35 b and | a | as small as possible. 6 (3 points) . Using (*) yet again, solve the congruence 35 x โก 2 mod 89....
View Full Document

## This note was uploaded on 10/31/2009 for the course STAT 131A taught by Professor Isber during the Spring '08 term at Berkeley.

Ask a homework question - tutors are online