Linear Congruences Homework

Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.

(1) Find all solutions to the linear congruence

(2) Find all solutions to the linear congruence

(3) Find all solutions to the linear congruence

(4) Find all solutions to the linear congruence

(5) Find all solutions to the linear congruence

(6) Find all solutions to the linear congruence

(7) Find all solutions to the linear congruence

(8) For which integers does the congruence have solutions? When there are solutions, how many incongruent solutions are there?

(9) Find an inverse modulo 13 of each of the following integers.

(10) Determine which integers where have an inverse modulo 14 and for each one find its inverse.

(11) Determine which integers where have an inverse modulo 30 and for each one find its inverse.

(12) Show that if is an inverse of modulo and is an inverse of modulo then is an inverse of modulo

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