Fundamental Theorem of Arithmetic 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 the prime factorization of

(2) Find all prime factors of

    (a)    (b)    (c)    (d)

(3) Show that all of the prime powers in the prime-power factorization of an integer are even if and only if is a perfect square.

(4) Show that if and are positive integers and then

(5) Show that if and then

(6) Find the least common multiple of each of the following pairs of integers:

    (a)    (b)    (c) 28, 35   (d) 111, 303   (e) 256, 5040   (f) 343, 999

(7) Find the greatest common divisor of each of the following pairs of integers:

(8) Find the least common multiple of each of the following pairs of integers:

(9) Show that if and are integers, then if and only if and

(10) If is a prime and is an integer with then

(11) Show that if is prime, is an integer, and is a positive integer such that then

(12) Show that is irrational.

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