Euler's Theorem 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 a reduced residue system modulo

(2) Find a reduced residue system modulo

(3) Find a reduced residue system modulo

(4) Find a reduced residue system modulo

(5) Find a reduced residue system modulo

(6) Find a reduced residue system modulo

(7) Show that if is a reduced residue system modulo where is a positive integer greater than 2, then

(8) Show that if and are positive integers with then

(9) Find the last digit in the decimal expansion of

(10) Find the last digit in the decimal expansion of

(11) Use Euler's theorem to find the least positive residue of modulo 35.

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