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 ≤ ≤ 40 [1b] 35x ≡ 15 (mod 64), 0 ≤ ≤ 64 [1c] 0 ≤ ≤ 17 • 19 = 323 �

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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 ≤ ≤ 40
[1b] 35x ≡ 15 (mod 64), 0 ≤ ≤ 64
[1c] 0 ≤ ≤ 17 • 19 = 323
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x ≡ 5 (mod 17)
x ≡ 11 (mod 19)
MATH 312 Final Exam
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[2] Given = 41 = 9. Solve each equation: (20 pts)
[2a] Find satisfying 1 ≤ < 41 and ≡ 1 ( 41)
[2b] Find satisfying 1 ≤ < 41 and ≡ 28 ( 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
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[4] Use [4*] to find the smallest positive integer solutions (if any) of each quadratic
congruence equation: (30 pts)
[4*] 12 ≡ 1 ( 11), 22 ≡ 4 ( 11), 32 ≡ 9 ( 11), 42 ≡ 5 ( 11),
52 ≡ 3 ( 11), 62 ≡ 3 ( 11), 72 ≡ 5 ( 11), 82 ≡ 9 ( 11),
92 ≡ 4 ( 11), 102 ≡ 1 ( 11)
[4a] Solve the congruence equation: 92 + 6 + 1 ≡ 0( 11), 0 ≤ ≤ 10