This is part of a homework Lesson; check the Lesson(s) online for due date and further instructions.
Math 323-2
Odd, Even, and Division Algorithm
Fall 2015
We have previously met the idea of odd and even numbers. In the context of the integers, the definitions
are:
• A number n is odd iff there is an integer k such that n = 2k + 1. (You can use 2k - 1 if you
prefer.)
• A number n is even iff there is an integer k such that n = 2k.
Two questions arise, the answers to which you may have thought were obvious:
• Is it possible for an integer to be BOTH even and odd?
• Are there any integers which are NEITHER even nor odd?
We will use the Division Algorithm to answer the second question in the way you know it should be
answered.
Solve the following problems:
0. Suppose m is an arbitrary integer and d = 2. Write down what the Division algorithm says in this
case (what happens when m is divided by 2?). After doing this, think about the remainder ... .
1. Suppose the remainder is 0. Is m even, odd, or neither? (Explain, of course.)
2. Suppose the remainder is 1. Is m even, odd, or neither? (Explain, of course.)
3. Are there any other possibilities?
4. Explain why every integer is either even or odd.
+1. (One can also conclude from the arguments above that an integer can’t be both even and odd, but
you are not asked to worry about this argument. [Note that this says “you are not asked to
worry about this”. It does not say “you are asked not to worry about this”.])
SO, from now on, you can assume:
[Partition property] Every integer is either even or odd, but not both.
In other words, if an integer is not even, then it is odd, and if it is not odd, then
it is even.
In addition, we have previously discussed, and you should be able to prove (but you are not asked to do
so here):
[Odd product property] The product of two odd numbers is odd.
[Even product property] The product of an even number and an integer (any integer) is even.
From now on, you can assume this.
Solve the following problems:
5. Prove that the square of an even integer is even (we’ve done it before, but do it again,.in this context).
6. Prove that the square of an odd integer is odd (ditto).
7. Prove that if n is an integer and n2 is even, then n is even. BE CAREFUL.
8. Prove that if ia is an integer and there exists an integer b such that a2 = 2b2, then a is even.
Oct 4, 2015 2:30 PM
This is part of a homework Lesson; check the Lesson(s) online for due date and further instructions.
Math 323-2
Irrational numbers
(Continued from previous sheet on Odd, Even, and Division Algorithm)
Fall 2015
As usual, you should USE PREVIOUS RESULTS, if relevant, when doing the following proof, for
example, Problems 0-8 on the preceding page. DON’T REPEAT WORK ALREADY DONE IN
PREVIOUS PROBLEMS in this problem set.
The point of these problems is to prove that !2 is not a rational number (see Problem 12). Don’t solve
them by using the fact that you know that !2 is an irrational number.
9. Prove that if a and b are integers such that !2 = a/b, then a has to be even.
(As usual, READ INSTRUCTIONS carefully This says “a has to be even”. It does not say “you
can choose a to be even”, which is obvious and trivial.)
10. Prove that, in the preceding problem, b also has to be even. (Optional hint: Write a explicitly as
an even number and solve for b -- but DON'T use !2 explicitly in your argument.)
11. Explain why, if we are given a rational number r and we write r = p/q for some integers p and q,
then we can choose p and q so that they are not both even.
12. Prove that !2 is not a rational number (or clearly explain why).
Oct 4, 2015 2:30 PM