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Use the extended Euclidean algorithm to find the greatest common divisor of 8,820 and 1,005 and express it as a linear combination of 8,820 and 1,005. Step 1: Find91 and r, so that 8,820 = 1,005 •91 + r1, where 0sr, < 1,005. Then r = 8,820 – 1,005 •91 = Step 2: Find q2 and r2 so that 1,005 = r1 · 92 + r2, where 0sr2 < r1: Then r2 = 1,005 – ( •92 = Step 3: Find 93 and r3 so that ri = r2· 93 + r3, where 0< r3 < r2. -( Then r3 = 93 =

Use the extended Euclidean algorithm to find the greatest common divisor of 8,820 and 1,005 and express it as a linear combination of 8,820 and 1,005. Step 1: Find91 and r, so that 8,820 = 1,005 •91 + r1, where 0sr, < 1,005. Then r = 8,820 – 1,005 •91 = Step 2: Find q2 and r2 so that 1,005 = r1 · 92 + r2, where 0sr2 < r1: Then r2 = 1,005 – ( •92 = Step 3: Find 93 and r3 so that ri = r2· 93 + r3, where 0< r3 < r2. -( Then r3 = 93 =

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.4: Prime Factors And Greatest Common Divisor
Problem 3E: In each part, find the greatest common divisor (a,b) and integers m and n such that (a,b)=am+bn....
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Use the extended Euclidean algorithm to find the greatest common divisor of 8,820 and 1,005 and express it as a linear combination of 8,820 and 1,005. Step 1: Find q, and r, so that 8,820 = 1,005 •q1 + r1, where 0<r, < 1,005. Then r, = 8,820 – 1,005 •91 = Step 2: Find 92 and r, so that 1,005 = r1: 92 + r2, where 0 s r2 < rị. Then r, = 1,005 – • 92 = Step 3: Find 93 and r3 so that ri = r2· 93 + r3, where 0<r3< r2. -(, Then r3 = •93 %3D Step 4: Find 94 and r4 so that r2 = r3· 94 + r4, where 0 <r4< r3. |-( Then r4 = 94 = Step 5: Find 95 and r5 so that r3 = r4: 95 + r5, where 0 s r5 < r4. Then r5 = • 95 Step 6: Conclude that gcd (8820, 1005) equals which of the following. gcd (8820, 1005) = r1 - r2 ·94 gcd (8820, 1005) = r4 - r5 93 gcd (8820, 1005) = r2 - r3 ·94 o gcd (8820, 1005) = r2 - r4· 95 o gcd (8820, 1005) = r3 - r4 • 95 Conclusion: Substitute numerical values backward through the preceding steps, simplifying the results for each step, until you have found numbers s and t so that gcd (8820, 1005) = 8,820s + 1,005t, where s = and t =
Transcribed Image Text:Use the extended Euclidean algorithm to find the greatest common divisor of 8,820 and 1,005 and express it as a linear combination of 8,820 and 1,005. Step 1: Find q, and r, so that 8,820 = 1,005 •q1 + r1, where 0<r, < 1,005. Then r, = 8,820 – 1,005 •91 = Step 2: Find 92 and r, so that 1,005 = r1: 92 + r2, where 0 s r2 < rị. Then r, = 1,005 – • 92 = Step 3: Find 93 and r3 so that ri = r2· 93 + r3, where 0<r3< r2. -(, Then r3 = •93 %3D Step 4: Find 94 and r4 so that r2 = r3· 94 + r4, where 0 <r4< r3. |-( Then r4 = 94 = Step 5: Find 95 and r5 so that r3 = r4: 95 + r5, where 0 s r5 < r4. Then r5 = • 95 Step 6: Conclude that gcd (8820, 1005) equals which of the following. gcd (8820, 1005) = r1 - r2 ·94 gcd (8820, 1005) = r4 - r5 93 gcd (8820, 1005) = r2 - r3 ·94 o gcd (8820, 1005) = r2 - r4· 95 o gcd (8820, 1005) = r3 - r4 • 95 Conclusion: Substitute numerical values backward through the preceding steps, simplifying the results for each step, until you have found numbers s and t so that gcd (8820, 1005) = 8,820s + 1,005t, where s = and t =
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