exam1sol - Math 465, Exam I (100 pts) Print Name: February...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 465, Exam I (100 pts) February 25, 2008 Print Name: Ae Ja Yee This exam is closed-book, closed-notes. Show all your work for full credit. Partial credit will be given based on what is written. You have 50 minutes to finish the exam. 1. (15 pts) Use the Euclidean algorithm to compute (171 , 30) and express (171 , 30) in the form 171 x + 30 y . Solution. Applying the Euclidean algorithm, we obtain 171 = 30 · 5 + 21 30 = 21 · 1 + 9 21 = 9 · 2 + 3 9 = 3 · 3 + 0 Therefore, (171 , 30) = 3. Working backwards through the equations, we obtain 3 = 21 - 9 · 2 = 21 - (30 - 21) · 2 = 21 · 3 - 30 · 2 = (171 - 30 · 5) · 3 - 30 · 2 = 171 · 3 - 30 · 17 Thus we have 3 = 171 · 3 + 30 · ( - 17). 2. Let Let a, b, c be integers. Prove that (a) (10 pts) if ( a, b ) = 1 and b | ac , then b | c ; (b) (10 pts) if ( a, b ) = 1, then ( a - b, 2 a + b ) = 1 or 3; (c) (10 pts) if 2 p - 1 is prime then p is prime. Solution.
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 3

exam1sol - Math 465, Exam I (100 pts) Print Name: February...

This preview shows document pages 1 - 2. Sign up to view the full document.

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