Please see attached file for full problem description.
2. Find the prime factorizations of (482,1687). Thus find the gcd and the lcm of the pair. Also find the gcd by Euclid's algorithm and then find the lcm from lcm(x,y)=xy/gcd(x,y).
3. Prove the following statment by induction. Use the headings Base, Inductive step, and By induction.
n3 - n is divisible by 6, for every natural number n > 1.
4. Show that the group (Z/16Z)× has 8 elements, and that each element has order 1, 2 or 4. Use the following definition of the order of an element in a finite group : The order of an element g in a finite group G is the least positive integer n such that gn=1.
9. In each of the following exercises, set S and a binary operation ∗ on S will be specified. In each case, determine whether ∗ is associative, whether ∗ has an identity element and, if the latter is the case, whether each element of S has an inverse (with regard to ∗), i.e. whether S is a group: (State reason for each)
STATE REASON WHY?
Euclid's algorithm is investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.