Introduction to 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) Show that each of the following congruences hold

(2) Determine whether each of the following pairs of integers is congruent modulo7.

(3) Show that if is an even integer, then and if is an odd integer then

(4) Show that if is an odd integer, then   

(5) Find the least nonnegative residue modulo 13 of each of the following integers:

(6) Find the least positive residue of modulo each of the following integers:

(7) Show that if and are integers such that and then

(8) Show that if and are integers such that and then

(9) Show that if and are integers such that such that then

(10) Construct a table for addition modulo 6.

(11) Construct a table for subtraction modulo 6.

(12) Construct a table for multiplication modulo 6.

(13) What can you conclude if where and are integers and is prime?

(14) Show that if is odd positive integer, then

(15) Show by mathematical induction that if is a positive integer,  then

(16) Show by mathematical induction that if is a positive integer,  then

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