Mathematics 5

Winter Term 2002

The World According to Mathematics

 

                                                                                    Dwight Lahr and David Rudel

 

Final Exam-Problem Solving

 

Due Saturday, March 9

 

 

Total Points: 75                                                Name:______________________________

 

 

On Saturday, March 9, before 10:00 a.m., hand in this sheet at Dwight Lahr's office, 410 Bradley with your solutions to the following problems. You should feel free to ask questions of either Dwight Lahr or David Rudel as you work on these problems. However, it is a violation of the honor principle to consult anyone else. You may use as resources only the manuscript and notes on the course website. You also may use a calculator. Be sure to show all of your work, including explanations of your reasoning. Good luck.

 

 

1.      A natural number is called perfect if it is equal to the sum of its divisors that are different from itself. For example, 6 is perfect because .

 

(a)     Show that 496 is perfect.

 

(b)         Euclid showed that  is prime if and only if  is perfect.

 

(i)                  Show that the fact that 496 is perfect follows from this theorem of Euclid’s.

(ii)                Using the theorem and other facts you know about primes, can you determine if  is perfect? Explain.

 

(c)     What is the sum of the reciprocals of all of the positive divisors of 6? of 28? (Is 28 perfect?)

 

(d)     What is the sum of the reciprocals of all of the positive divisors of 496?

 

(e)     From parts (c) and (d), what conjecture can you make about the sum of the reciprocals of all of the positive divisors of a perfect number?

                                                                                                [15 points]

 

 

2.      As an employee of a bookstore, you are asked by a customer to order a book known only by its ISBN, 8 - 401 - 32424 - X. After applying the ISBN algorithm to this number, answer the following questions (You may use the ISBN Maple worksheet.):

 

(a)     Why is it that the number is not a valid ISBN?

 

(b)     Use the techniques you know for correcting common errors in transmitted ISBN's to develop a list of possible corrections.

                                                                                                                        [6 points]

           

 

3.      Suppose n is a positive integer with prime power factorization n=pqr.

 

(a)     Give a formula for f(n) in terms of p, q, and r, where f is the Euler f-function.

 

(b)     Find f(30) using the formula you derived in part a.

 

(c)     List all the numbers less than 30 and relatively prime to 30.  Use this list to verify that the answer you got in part (b) is correct.

                                                                                                               [9 points]         

 

4.      There is a Maple worksheet called "FinalMsg.mws" that you should download from the "Take-Home Exams" section of the Math 5 website. The worksheet contains a message encoded according to our conventions for encoding/decoding, but in a code that you must break. Decrypt the message, and hand in a copy of the last two worksheets you use to obtain the message in plain English. The decoded message should appear in the last printout. [Note: you will need to use the Maple command "ifactor()" at some point in your work.]

 

         Note 1: Remember to activate any cell with the plus sign. (Open the section by clicking the plus sign. Then place the cursor in the cell and type the Enter key.)

 

         Note 2: If Maple will not print your decoded message, there are several possible work-arounds. For example, you can make a screen snap and print that. (On a Mac, you do this by clicking command-shift-3 and then printing the Picture using SimpleText.) Or you can highlight your blue output and copy and paste it into a Word document and print that.                                                                                                                                                                                                      [6 points]

 

5.      Let G be D4. (D4 is the group on {a,b} with the relations {a4=e, b2=e, ab=ba3}).  Let H be V (V is the group on {c,d} with the relations {c2=e,d2=e, cd=dc}).  Let f be a map that takes G to H such that  f(a)=c and  f(b)=cd.

            i) Verify that this map is valid.

            ii) Calculate f(a3b).

            iii) Calculate f(a-1).  [here "a-1" means "a hat"].

            iv) What is the kernel of this map?

            v) What is the image of this map?

vi) Make a new group by taking G and imposing new relations on it corresponding to its kernel.  What are the elements in the new group?

vii) What does the "big theorem" say regarding this new group?  Can you show the correspondence?

[12 points]

 

6.      Consider a triangular pyramid.  That is, a pyramid whose base is a triangle.  All the triangular faces (the bottom and all the sides) are equilateral triangles.  Consider all the things that can be done to the pyramid that place it back in the same place (with its corners possibly shifted).  We consider two things the same if they do the same thing to the corners.  For example, one thing that could be done is to rotate the pyramid's base but keep the same corner in the air.  Thus, if you assigned the number "1" to the tip that was sticking up and assigned the number "2" to the corner on the base that faces north, and assigned the number "3" to the corner facing southeast and "4" to the corner facing southwest, then rotating the pyramid one third turn clockwise (viewing it from the top) would keep the tip marked with a "1" in the same place.  It would take "2" to the place where "3" used to be.  It would take "3" to the place where "4" used to be, and it would take "4" to the place where "2" used to be.  Rotating it the other way would do the opposite: "2" would go to "4," "4" would go to "3," and "3" would go to "2."  Thus, if you rotated it twice clockwise it would be the same as rotating it once counterclockwise because the corners would wind up in the same position.

Model the group of actions as a group on a set of letters with relations, just as we did on class the first Friday of group theory.

i)    State what your group is in terms of what letters it uses and what the relations are.

ii)   What are the elements in your group? [Example, if I were doing Z4 I would say [e], [a], [a2], and [a3] if I were thinking of the group as a group on {a} with the relation {a4=e}.]

iii) Find a subgroup that has exactly 3 elements in it. Remember to show that it actually is a subgroup.

iv)  Is this group a kernel? Why or why not?

v)   Find a subgroup that has exactly 4 elements in it.  Show that it is a subgroup.

vi) Is this group a kernel?  Why or why not?

[15 points]

7.      Recall that I emphasized that the idea of a kernel is that it is what "we don't care about."  In class we discussed the "symmetries of a square," [We call it D4, the "4" is because a square has that many sides] that is all the movements one could do to a square where two movements are the same if they take the corners to the same places.  We found that this group could be thought of as a group on {a,b} with the added relations that {a2=e, b4=e, and ab=b3a}.  Here "a" represents a flip (say, vertically), and "b" represents a rotation (say, clockwise).

         Now, pretend that you color two opposite sides of the square blue and the other two sides red.  Now, any movement you do will either take blue to blue and red to red or will take red to blue and blue to red.  Thus, each possible movement either changes the colors (taking red to blue and blue to red) or preserves the colors (taking blue sides to blue sides and red sides to red sides).

         Pretend that all you care about is whether a movement  is a changer or a preserver.  Make a new group, call it J, whose only elements are {C, P}.  [Note, these are not the letters, but the elements]. 

i) Make a multiplication table for J. [We think of C*P as "Do something that changes the colors and then do something (after that) which preserves them.  So, if doing something that changes the colors and then doing something that preserves the (changed) colors overall changes the colors we would say C*P=C.  Your table should be small…after all the group has only 2 elements in it.]

ii) Make a map into this group where all the elements of the first group (All the movements) are mapped to their result.  That is, elements that change the colors go to "C" and elements that preserve the colors go to "P."

iii) Show this is a valid map.

iv) What is the kernel of your map?

v) Why does it make sense for this to be the kernel when we think of the kernel as "things we don't care about."?

vi) Make a new group by imposing extra relations corresponding to the kernel.  What are the elements in your new group?

[8 points]

8.      Let G  be a group that has the property that X*Y=Y*X for all X,Y in G.  Show that every subgroup of G is a kernel.

                                                                                                            [4 points]

 

BONUS

All of these are optional

 

1.      In problem 6 one of the subgroups mentioned is, in fact, a kernel.  Give an example of a map for which it would be the kernel.

[3 points]

 

2.      Show that if a group is finite, for each letter there is a power of that letter that is equal to the "hat" of that letter.  For example, if G is a group on {a,b,c}, and G is finite, then there is some power n  such that an=a-1  [where the "-1" means "hat"].

[3 points]

 

3.      Show that if there are n  elements in any (one)  pre-image, there are also n  elements in the kernel.

[3 points]

 

 

4.      Suppose n is a positive integer. If p is a prime that divides n (evenly), then prove that f(pn) = p f (n).                                

                                                                                                                        [3 points]