Number theory proofs problem set

Problem 1

(a) Prove or Disprove: For all ??, ??, ?? ? Z

+, if ??|????, then ??|?? or ??|??.

Note! Z

+ is the set of all positive integers.

(b) Prove or Disprove: For all ??, ?? ? Z=2 where ??|??, if ??=?? (mod ??) (where ??, ?? ? Z), then

??=?? (mod ??). Note! Z is the set of all integers. Z=2 is the set of all integers = 2.

Problem 2

(a) Prove or Disprove: For all integers ?? ? Z

*

, ??

2 + ?? + 41 is a prime number.

Note! Z

*

is the set of all positive integers including 0.

(b) Prove or Disprove: 2 is the only even prime number.

Problem 3

(a) Prove or Disprove: For all ?? ? Z

+, ?? and ?? + 1 are relatively prime.

(b) Prove or Disprove: For all ?? ? Z

+, ?? and ?? + 2 are relatively prime.

(c) Prove or Disprove: For all ?? ? Z

+, if ?? is odd, then ?? and ?? + 2 are relatively prime.

(d) Prove or Disprove: For all ??, ??, ?? ? Z=2 if ??|?? and ??|??, then ?? + ?? and ?? are not relatively

prime.

Note! Z

+ is the set of all positive integers. Z=2 is the set of all integers = 2.

Problem 4 Least Common Multiple (lcm)

Say we have the integer ?? = 204,459,408,000, and suppose we know that ?? = lcm(a, b) and

?? = 709,928,500. How many possible values of ?? are there if ?? is a positive integer?

(Hint: All of ?? and ??’s prime factors are = 29.)

Problem 5 Greatest Common Divisor (gcd)

Use the Extended Euclidean Algorithm to express gcd (225, 431) as a linear combination of 225

and 431.

(Hint: There’s a great YouTube video that can help with this if you search “Extended Euclidean

Algorithm”)

Problem 6

Consider a discrete random variable, ??, with the following probability distribution:

What is the expected value of ???

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