Numerical analysis homework help
MATH 312 Final Exam Past Due Date Final Exam Will Not Be accepted
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(Show all work.) [1] Find the small positive integer solution of each problem: (38 pts)
(Hint: you may find any integer solution first, then add/subtract a multiple of n)
[1a] 7x โก 6 (mod 41), 0 โค ๐ฅ๐ฅ โค 40
[1b] 35x โก 15 (mod 64), 0 โค ๐ฅ๐ฅ โค 64
[1c] 0 โค ๐ฅ๐ฅ โค 17 โข 19 = 323
๏ฟฝ x โก 5 (mod 17) x โก 11 (mod 19)
Due on Sunday, December 06, 2020 Print your name: _________________________________.
Page 1 of 3
(Show all work.) [1] Find the small positive integer solution of each problem: (38 pts)
(Hint: you may find any integer solution first, then add/subtract a multiple of n)
[1a] 7x โก 6 (mod 41), 0 โค ๐ฅ๐ฅ โค 40
[1b] 35x โก 15 (mod 64), 0 โค ๐ฅ๐ฅ โค 64
[1c] 0 โค ๐ฅ๐ฅ โค 17 โข 19 = 323
๏ฟฝ x โก 5 (mod 17) x โก 11 (mod 19)
MATH 312 Final Exam Past Due Date Final Exam Will Not Be accepted
Due on Sunday, December 06, 2020 Print your name: _________________________________.
Page 2 of 3
[2] Given ๐๐ = 41 ๐๐๐๐๐๐ ๐๐ = 9. Solve each equation: (20 pts) [2a] Find ๐๐ satisfying 1 โค ๐๐ < 41 and ๐๐๐๐ โก 1 (๐๐๐๐๐๐ 41)
[2b] Find ๐๐ satisfying 1 โค ๐๐ < 41 and ๐๐๐๐ โก 28 (๐๐๐๐๐๐ 41) (Use [2a] is easier.)
[3] Find the set of quadratic residues and the set of nonresidues of p = 17. 12 pts
MATH 312 Final Exam Past Due Date Final Exam Will Not Be accepted
Due on Sunday, December 06, 2020 Print your name: _________________________________.
Page 3 of 3
[4] Use [4*] to find the smallest positive integer solutions (if any) of each quadratic congruence equation: (30 pts)
[4*] 12 โก 1 (๐๐๐๐๐๐ 11), 22 โก 4 (๐๐๐๐๐๐ 11), 32 โก 9 (๐๐๐๐๐๐ 11), 42 โก 5 (๐๐๐๐๐๐ 11),
52 โก 3 (๐๐๐๐๐๐ 11), 62 โก 3 (๐๐๐๐๐๐ 11), 72 โก 5 (๐๐๐๐๐๐ 11), 82 โก 9 (๐๐๐๐๐๐ 11),
92 โก 4 (๐๐๐๐๐๐ 11), 102 โก 1 (๐๐๐๐๐๐ 11)
[4a] Solve the congruence equation: 9๐ฅ๐ฅ2 + 6๐ฅ๐ฅ + 1 โก 0(๐๐๐๐๐๐ 11), 0 โค ๐ฅ๐ฅ โค 10
[4b] Solve the congruence equation: 5๐ฅ๐ฅ2 + 3๐ฅ๐ฅ + 2 โก 0 (๐๐๐๐๐๐ 11), 0 โค ๐ฅ๐ฅ โค 10
[4c] Solve the congruence equation: 3๐ฅ๐ฅ2 โ 4๐ฅ๐ฅ + 1 โก 0 (๐๐๐๐๐๐ 11), 0 โค ๐ฅ๐ฅ โค 10