soln4 - MATH 135 Algebra, Solutions to Assignment 4 1: For...

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MATH 135 Algebra, Solutions to Assignment 4 1: For each of the following pairs ( a,b ), find integers q and r with 0 r < | b | such that a = bq + r . (a) a = 753, b = 21 Solution: Using long division (or using a calculator) we find that 753 = 21 · 35 + 18, so q = 35 and r = 18. (b) a = - 5124, b = 316 Solution: Using long division (or using a calculator) we find that 5124 = 316 · 16 + 68, and so we have - 5124 = 316( - 16) - 68 = 316( - 17) + 316 - 68 = 316( - 17) + 248. Thus q = - 17 and r = 248. (c) a = 4137, b = - 152 Solution: Using long division (or using a calculator) we find that 4137 = 152 · 27 + 33 = ( - 152)( - 27) + 33, and so q = - 27 and r = 33. 2: For each of the following pairs ( a,b ), find gcd( a,b ). (a) a = 78, b = 34 Solution: Applying the Euclidean Algorithm gives 78 = 2 · 34 + 10 34 = 3 · 10 + 4 10 = 2 · 4 + 2 4 = 2 · 2 + 0 and so gcd( a,b ) = 2. (b) a = 456, b = 1273 Solution: The Euclidean Algorithm gives 1273 = 2 · 456 + 361 456 = 1 · 361 + 95 361 = 3 · 95 + 76 95 = 1 · 76 + 19 76 = 4 · 19 + 0 and so gcd( a,b ) = gcd(1273 , 456) = 19. (c) a = - 1205, b = 2501 Solution: The Euclidean Algorithm gives 2501 = 2 · 1205 + 91 1205 = 13 · 91 + 22 91 = 4 · 22 + 3 22 = 7 · 3 + 1 3 = 3 · 1 + 0 so we have gcd( a,b ) = gcd(2501 , 1205) = 1.
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3: For each of the following pairs ( a,b ), find d = gcd( a,b ) then find integers s and t such that as + bt = d . (a) a = 60, b = 35 Solution: The Euclidean Algorithm gives 60 = 1 · 35 + 25 , 35 = 1 · 25 + 10 , 25 = 2 · 10 + 5 , 10 = 2 · 5 + 0 so we have d = gcd( a,b ) = 5. Back-Substitution then gives rise to the sequence 1 , - 2 , 3 , - 5 so we have 60 · 3 - 35 · 5 = 5 and can take s = 3 and t = - 5. Alternatively, the Extended Euclidean Algorithm gives rise to the table 1 0 60 0 1 35 1 - 1 25 - 1 2 10 3 - 5 5 so we have 60 · 3
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soln4 - MATH 135 Algebra, Solutions to Assignment 4 1: For...

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