Number theory proofs problem set

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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