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exam1sol

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

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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. (a) Since ( a, b ) = 1, 1 = ax + by for some x, y . So c = acx + bcy. Since b | ac , b | c . (b) Let a - b = m and 2 a + b = n . Then a = m + n 3 , b = n - 2 m 3 . Since ( a, b ) = 1, for some x, y , 1 = ax + by. 1

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Then 3 = ( m + n ) x + ( n - 2 m ) y = m ( x - 2 y ) + n ( x + y ) , which shows that ( m, n ) | 3. Therefore, ( m - n ) = 1 or 3.
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