Math 160 Mathematical Problems and Games, Fall 2002
Problem Set 7: hand in one problem Tuesday, November 5.

As always, think about all of the problems.  Then choose one to write up and hand in.  You can also revise and hand in a problem from a previous week.  Be sure to hand in your original solution with the revision.

1.  Once upon a time, there was a student who ran out of money.  He sent the following note home to his parents, who were both math teachers:

 SEND
+MORE
MONEY

Each letter stands for a different digit 0-9.  0 never appears at the beginning of a number.  How much money ($MON.EY) was the student asking for?

2.  Long division!

      CECI
QUI|TROUVE
    ---
     ---
     ---
      ---
      --E
       ---
       ---
         0

Again, each letter stands for a different digit 0-9.  Which letters represent which numbers?  ("Qui trouve ceci?" is French for "Who finds this?").

3.  a) Show that the last digit of the square of any integer must be 0, 1, 4, 5, 6, or 9.
    b)  What digits can be the last digit of the cube of a positive integer?
    c)  What digits can be the last digit of the fourth power of a positive integer?
   d)  Show that the next to last digit of the square of an integer is odd if and only if the last digit is 6.
   e)  Part of a document from Renaissance Florence was found, giving the area in square feet of several different buildings.  The fragment that was recovered only has the ends of the numbers on it, but each building had an area of at least 100,000 square feet.  Which building was perfectly square?

....46  Uffizi
...235  Medici chapel
...168  Pitti Palace
....44  Ospedali
..4327  Duomo

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Remember that Tuesday, November 5 is election day!  If you're eligible to vote, be sure to do so.  If you want information about candidates, try the League of Women Voters  at www.dnet.org or Project Vote Smart at www.vote-smart.org .