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2007exam1sol

# 2007exam1sol - Math 465 Exam I(100 pts Print Name Ae Ja Yee...

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Math 465, Exam I (100 pts) February 23, 2007 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. (10 pts) Use the Euclidean algorithm to compute (300 , 230) and express (300 , 230) in the form 300 x + 230 y . Solution. Applying the Euclidean algorithm, we obtain 300 = 230 · 1 + 70 230 = 70 · 3 + 20 70 = 20 · 3 + 10 20 = 10 · 2 Therefore, (300 , 230) = 10. Working backwards through the equations, we obtain 10 = 70 - 20 · 3 = 70 - (230 - 70 · 3) · 3 = 70 · 10 - 230 · 3 = (300 - 230) · 10 - 230 · 3 = 300 · 10 - 230 · 13 Thus we have 10 = 300 · 10 + 230 · ( - 13). 2. Let a, b, c Z . Prove that (a) (10 pts) if a | b and a | c , then a | ( bx + cy ) for all x, y Z ; (b) (10 pts) if a is composite then 2 a - 1 is composite. Solution. (a) Since a | b and a | c , b = au and c = av for some u, v Z . So bx + cy = aux + avy = a ( ux + vy ) , which shows that

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2007exam1sol - Math 465 Exam I(100 pts Print Name Ae Ja Yee...

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