Problem Set One ME 581 - University of Arizona



SED ME 581 Advanced Topics in Algebra for Teachers: Problem Set 1

P1: Prove that the sum of the first n positive integers is n(n + 1)/2.

P2: Prove that 22n – 1 is divisible by 3 for every positive integer n.

P3: Prove that 11n – 4n is divisible by 7 for n = 1, 2, … .

P4: Prove that the sum of the squares of the first n integers is n(n + 1)(2n + 1)/6.

P5: My post office sells only 5-cent and 9-cent stamps. Show that any amount of postage greater than 35¢ can be paid using only these stamps.

Modular Arithmetic

Modular arithmetic is often called clock arithmetic, because it is like doing addition or multiplication using a clock. That is, we can say that 7 hours after 8 o’clock, it will not be 15 o’clock (unless you are using military time), it will instead be 3 o’clock. So, in clock arithmetic, 8 + 7 = 3. Thus we say that 15 mod 12 = 3. Clock arithmetic partitions that integers into 12 distinct classes, one for each hour on the clock. Thus 0, 12, 24, 36,-12, and all multiples of 12 belong to one class, all integers that are one more than a multiple of 12 belong to another class, and so forth, so that you have twelve distinct classes.

We can use any positive integer greater than one as the modulus. The integers mod 2 partition the integers into two distinct classes, those that are even and those that are odd. We write Z2 to indicate the integers mod 2, and Zp to indicate the integers mod p, any prime number.

P6: Complete the table (called a Cayley Table after Arthur Cayley, 1854) to show the sums and products for addition and multiplication mod 5. Always first use the number in the right column, then follow with the number along the top.

+ 1 2 3 4 5 x 1 2 3 4 5

1. 1

2. 2

3. 3

4. 4

5. 5

P7: What is the identity element for addition mod 5? The identity element is a “do nothing” element that leaves each element in the set alone. What is the additive inverse for each element in the set? The additive inverse for each element will “undo” what the element did, and get it back to the identity.

P8: What is the identity element for multiplication mod 5? The identity element is a “do nothing” element that leaves each element in the set alone. What is the multiplicative inverse for each element in the set? The multiplicative inverse for each element will “undo” what the element did, and get it back to the identity.

P9: If two numbers a and b are congruent mod m, they have the same remainder when divided by m, and we write a [pic] b mod m. Show that if a [pic] b mod m, then m divides a – b.

The Definition of Group

A mathematical system consists of a set of elements with one or more operations defined on that set of elements, and some rules governing the operations on the elements. The group is perhaps the simplest mathematical system, and yet the group is one of the most important and useful mathematical systems we know.

A group is a mathematical system consisting of a set of elements, S, one binary operation, *, and the following axioms (or rules).

0) The set S is closed under the operation *. That means that the result of using this operation on any two elements in the set will give an element in the set.

In symbols: [pic]a, b, in S, a*b = c, and c is also in S.

1) The operation * is associative. That means that that the associative law holds when you perform the binary operation on three or more elements.

In symbols: [pic]a, b, c, in S, (a*b) * c = a*(b * c).

2) There is a unique identity element, e, in the set. That means that there exists a “do nothing” element that leaves each element in the set alone.

In symbols, [pic]an e in S, such that [pic]a in S, ea = ae = a.

3) For each element in S, there is an inverse element. That means that each element has an inverse that will “undo” what it does and get it back to the identity.

In symbols, [pic]a in S, [pic] an a-1 in S, such that a-1a = aa-1 = e.

The Parade Group

Here is an example of a group. The set of elements is the set containing the four parade commands: left face (L), right face (R), about face (A), and stand as you were (S). The operation is “followed by”, which we will designate as F.

P10: Make a Cayley Table to show the results of each command followed by other commands.

F L R A S

L

R

A

S

P11: Prove that the “Parade Group” really is a group. That is, show that the group axioms hold for the four commands and the operation “followed by”.

P12: Look back at the tables you made for P9. Do the integers mod 5 form a group under addition? Explain. Do the integers mod 5 form a group under multiplication? Explain.

P13: Make addition and multiplication tables for the integers mod 2, 3, 4, 6, 7, 8, 9, and 10. Which ones form groups for addition? Which ones form groups for multiplication? Make a conjecture about which values of n will work for addition, and which will work for multiplication.

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