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 ≡ 6 (mod 41), 0 ≤ 𝑥𝑥 ≤ 40
 
[1b] 35x ≡ 15 (mod 64), 0 ≤ 𝑥𝑥 ≤ 64
[1c] 0 ≤ 𝑥𝑥 ≤ 17 • 19 = 323
� x ≡ 5 (mod 17) x ≡ 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: _________________________________.
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[2] Given 𝑝𝑝 = 41 𝑎𝑎𝑎𝑎𝑎𝑎 𝑔𝑔 = 9. Solve each equation: (20 pts) [2a] Find 𝑎𝑎 satisfying 1 ≤ 𝑎𝑎 < 41 and 𝑎𝑎𝑔𝑔 ≡ 1 (𝑚𝑚𝑚𝑚𝑎𝑎 41)
 
[2b] Find 𝑚𝑚 satisfying 1 ≤ 𝑚𝑚 < 41 and 𝑚𝑚𝑔𝑔 ≡ 28 (𝑚𝑚𝑚𝑚𝑎𝑎 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 ≡ 1 (𝑚𝑚𝑚𝑚𝑎𝑎 11), 22 ≡ 4 (𝑚𝑚𝑚𝑚𝑎𝑎 11), 32 ≡ 9 (𝑚𝑚𝑚𝑚𝑎𝑎 11), 42 ≡ 5 (𝑚𝑚𝑚𝑚𝑎𝑎 11),
52 ≡ 3 (𝑚𝑚𝑚𝑚𝑎𝑎 11), 62 ≡ 3 (𝑚𝑚𝑚𝑚𝑎𝑎 11), 72 ≡ 5 (𝑚𝑚𝑚𝑚𝑎𝑎 11), 82 ≡ 9 (𝑚𝑚𝑚𝑚𝑎𝑎 11),
92 ≡ 4 (𝑚𝑚𝑚𝑚𝑎𝑎 11), 102 ≡ 1 (𝑚𝑚𝑚𝑚𝑎𝑎 11)
[4a] Solve the congruence equation: 9𝑥𝑥2 + 6𝑥𝑥 + 1 ≡ 0(𝑚𝑚𝑚𝑚𝑎𝑎 11), 0 ≤ 𝑥𝑥 ≤ 10
 
 
 
 
 
 
[4b] Solve the congruence equation: 5𝑥𝑥2 + 3𝑥𝑥 + 2 ≡ 0 (𝑚𝑚𝑚𝑚𝑎𝑎 11), 0 ≤ 𝑥𝑥 ≤ 10
 
 
 
 
 
[4c] Solve the congruence equation: 3𝑥𝑥2 − 4𝑥𝑥 + 1 ≡ 0 (𝑚𝑚𝑚𝑚𝑎𝑎 11), 0 ≤ 𝑥𝑥 ≤ 10