Midterm2Solutions

Midterm2Solutions - Faculty of Mathematics University of...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Faculty of Mathematics University of Waterloo MATH 135 MIDTERM EXAM #2 Fall 2007 Monday 12 November 2007 19:00 20:15 Solutions 1. In each part of this problem, full marks will be given if the correct answer is written in the box. If your answer is incorrect, your work will be assessed for part marks. (a) Convert (2345) 6 to base 10. [3] Answer: (2345) 5 = 569 Solution By definition, (2345) 6 = 2(6 3 ) + 3(6 2 ) + 4(6) + 5 = 2(216) + 3(36) + 4(6) + 5 = 569 so (2345) 6 = (569) 10 . (b) Convert (2007) 10 to base 12, using A and B to represent the digits 10 and 11, respectively. [3] Answer: (2007) 10 = (11 B 3) 12 Solution Using the conversion algorithm, 2007 = 167(12) + 3 167 = 13(12) + 11 13 = 1(12) + 1 1 = 0(12) + 1 Therefore, (2007) 10 = (11 B 3) 12 , since B represents the digit 11. (c) Determine the remainder when 2 34 56 52 + 3 19 is divided by 17. [3] Answer: 11 Solution Since 17 is prime and none of 2, 56 or 3 is divisible by 17, then 2 16 56 16 3 16 1 (mod 17) by Fermats Little Theorem. MATH 135, Midterm #2 Solutions Page 2 of 5 Therefore, 2 34 56 52 + 3 19 (2 16 ) 2 2 2 (56 16 ) 3 56 4 + (3 16 )3 3 (mod 17) 1 2 (4)1 3 5 4 + 1(27) (mod 17) (since 56 5 (mod 17)) 4(625) + 27 (mod 17) 2527 (mod 17) 11 (mod 17) Therefore, the remainder is 11. (d) Determine the number of congruence classes in Z 18 that are solutions to the equation [3] [12][ x ] = [5]....
View Full Document

Page1 / 5

Midterm2Solutions - Faculty of Mathematics University of...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online