The Babylonians used base sixty for their arithmetic instead of base ten like we use. They had a good reason for this: the ancient Babylonians had sixty fingers! (pause for laugh] In this Big Problem we will explore the advantages and disadvantages of the ancient Babylonian system.
(1) Make a list of all the divisors of sixty. It should be a long list!
(2) Each divisor generates a subgroup of Zso under addition. Write out the operation table for one of these subgroups.
(3) Write out the operation table for another of these subgroups.
(4) What are the elements of the multiplicative group of integers modulo 60? HINT: Take a look at Definition 11.4.17 in your textbook.
(5) Let's compare the Babylonian system with another system. Make a list of all the divisors of one hundred. How does this compare to your answer to part 1?
(6) Each divisor generates a subgroup of Z100 under addition. Write out the operation table for one of these subgroups.
(7) Write out the operation table for another of these subgroups.
(8) What are the elements of the multiplicative group of integers modulo 100? Hint: It will probably be easier to start by finding the integers that are not in this group.
(9) Compare your answers to one and five, and four and eight. What do you notice? Why do you think that happened?
(10) If you were in charge of Babylonian money, what denomination of coins would you make, and why? For example, we use pennies (one cent) nickels (five cents), dimes(ten cents), and quarters (twenty-five cents).