Question 7(a) For integers a, b, q, r, prove that if a = bq + r then (a, b) = (b, r).(b) Determine (1000, 2025) by using the Euclidean algorithm. Then express the answer as alinear combination of 1000 and 2025. Show all your working.(c) Suppose that G== (a), a e, and a³ = e. Construct a Cayley table for the group (G,.).(d) Determine the right cosets of ((1 2), [1]) € S3 x Z₂.(e) Find the subgroups of Z24.

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Determine the groups of quadratic residues Qn for all n in the range 10 ≤ n ≤ 20.
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A student is tracing the following algorithm. The function INT gives the integer part of any number, eg INT(2.3) = 2 and INT (6.7) = 6. Line 10 Input A, B Line 20 Let C = INT(A ÷ B) Line 30 Let D = B × C Line 40 Let E = A – D Line 50 If E = 0 then go to line 90 Line 60 Let A = B Line 70 Let B = E Line 80 Go to line 20 Line 90 Print B Line 100 Stop Trace the algorithm in the case where the input values are: (a) (i) A = 36 and B = 16; (ii) A = 11 and B = 7. (b) State the purpose of the algorithm.
Introduction To Discrete Mathematics: 1. Use Euclid’s Algorithm to calculate gcd(235, 145).2. Find a solution to 235x + 145y = 5.
(a) For integers a, b, q, r, prove that if a = bq + r then (a, b) = (b, r). (b) Determine (1000, 2025) by using the Euclidean algorithm. Then express the answer as a linear combination of 1000 and 2025. Show all your working. (c) Suppose that G = (a), a ‡e, and a5 : = e. Construct a Cayley table for the group (G,.). (d) Determine the right cosets of ((1 2), [1]) € S3 × Zi₂. (e) Find the subgroups of Z24.
Please include justifications for each step.
Theorem 1.2. Division Algorithm: Given integers a and b, with b>0, there exist unique integers q and r satisfying a = bq + r 0
Sort 6, 2, 5, 4, 3 in increasing order using the following algorithm sort(a(1),..., a(n): real numbers with n ≥ 2) for i:=1 to n-1 for j:=1 to n-i if a(j) > a(j+1) then interchange a(j) and a(j+1) {a(1),...,a(n) is in increasing order} Which of the following steps do not attained during the procedure? 1. 2, 5, 6, 4, 3 II. 2, 6, 4, 5, 3 III. 2, 4, 3, 5, 6 IV. 2, 6, 5, 4, 3 V. 2, 4, 5, 6, 3 II and V I and II III and IV only V

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