Question

For each of the following congruences if there is a solution, express the solution in the form x ≡  some_number  (mod some_modulus), e.g. x ≡ 6 (mod 9). To standardize answers,  some_number should always be a value in the range {0, 1, 2, ..., some_modulus -1}. For example x ≡ 5 (mod 8) is OK but x ≡ 13 (mod 8) is not.

If there is no solution say "No solution". You don't have to show work for any of the problems. Type your answers in the Write Submission box. For example

a) x ≡ 4 (mod 17)

b) x ≡ 5 (mod 21)

c) No solution

etc.

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Here are the problems.

a) 5x - 7 = 11 (mod 17)

b) 6x + 12 = 2 (mod 21)

c) 14x + 5 = 9 (mod 20)

d) 12x + 12 = 12 (mod 12)

e) (-7)x + 5 = 2 (mod 30)

f) 21x - 13 = 7 (mod 34)

g) 20x + 17 = 1 (mod 28)

h) -x + 13 = 28 (mod 2)

i) 1,000,000,000x + 82 = 5 (mod 9)

j) (5⁴⁴⁴¹⁵)x + 7³²⁶ = 4⁸⁸⁶⁷ (mod 17)

Answer 1

(a) 5x = 18 (mod 17)

So, 5x = 1 (mod 17)

So, 7•5x = 7 (mod 17)

So, 35x = 7 (mod 17)

So, x = 7 (mod 17)

(b) 6x = -10 = 9 (mod 21)

So, 2x = 3 (mod 7)

So, 8x = 12 = 5 (mod 7)

So, x = 5 (mod 7)

(c) 14x = 4 (mod 20)

So, 7x = 2 (mod 10)

So, 21x = 6 (mod 10)

So, x = 6 (mod 10)

(d) 12x = 0 (mod 12)

So, x can be any integer.

(e) (-7)x = -3 (mod 30)

So, 7x = 3 (mod 30)

So, 13•7x = 39 (mod 30)

So, 91x = 9 (mod 30)

So, x = 9 (mod 30)

(f) 21x = 20 (mod 34)

5•21x = 100 (mod 34)

So, 105x = 32 (mod 34)

So, 3x = -2 (mod 34)

So, 11•3x = 11(-2) (mod 34)

So, 33x = -22 (mod 34)

So, - x = -22 (mod 34)

So, x = 22 (mod 34)

(h) -x = 15 (mod 2)

So, x = -15 (mod 2)

So, x = 1 (mod 2)

(i) 1000000000x = -77 (mod 9)

So, x = -5 (mod 9)

So, x = 4 (mod 9)

(j) 544415x = 41541 (mod 17)

So, 7x = 10 (mod 17)

So, 5•7x = 50 (mod 17)

So, 35x = 16 (mod 17)

So, x = 16 (mod 17)