CSCI 2824: Lecture 7In this lecture we will cover:
Topics covered: Chapter 2.2 of the book. Proving Properties of NumbersLet us start with divisibility. An integer is divisible by a non-zero integer , if can be written as for some integers and . In logic, we write: The following concepts are all equivalent:
Divisibility by has a special name: If is divisible by , we call it even, and other wise we call it odd. Theorem -1: If are divisible by then is divisible by . Proof
For your convenience, we split the proof step by step to make the flow of reasoning steps clear.
Important Note
Whenever in a proof, a number can be written as , we conclude that is divisible by . Similarly, if you are told that a number is divisible by , it makes sense to write it as for some . Theorem-2 If is divisible by and by then is divisible by . Proof
Proofs on Consecutive NumbersTheorem-3 The product of two consecutive numbers is always even. Proof
Proofs by Case-SplittingA number is a perfect square if for some . Examples of perfect squares include and so on. Theorem-4 If a perfect square is even, then it is divisible by . Proof
Therefore, we conclude that must be divisible by based on case 2, which is the only case that can happen. QED. There are other ways of proving this theorem that we will revisit when we study proofs by contradiction. Modular ArithmeticWe have been using the “mod” operator so far in some proofs. Let us study the properties of the “mod” operator in more detail. Modulo Operator
Let be some integer. For a integer say that iff can be written as where is the quotient and is the remainder when is divided by . Let us look at examples. Example
We write . Since . As a convention, the result of a modulo operation is always between and . Also, we always apply over positive, but can be positive or negative What is ? We write . Therefore . Note:
Odd number: A number is odd iff . We can distribute over . Secondly, Let us prove these facts: Theorem: Suppose number can be written as for . It follows that . Proof
Let . Therefore for some . Note that . Therefore can be written as for . We conclude therefore that is the quotient when is divided by and is the remainder. Therefore, QED. Theorem: For any integers and integer , the following statement is true: Proof
Let us assume and . Therefore, we may write as for some . We also write as for some . Therefore, . QED. Similarly, we can prove that modulo operator distributes over multiplication. Theorem For any integers , and natural number , the following statement is true: Proof
Let and , where and . We can write and . Therefore, using the distributivity of modulo operator over addition (proved in the previous theorem), Therefore, This shows that (QED) Computing moduloLet us try some interesting problems involving modulo arithmetic. Example 1For instance, let us try and compute for various values of .
As you can see the pattern that emerges is
Example 2We know is a really large number. The question is very simple. Suppose we wrote in hexadecimal, what would the last digit be? In general, the last digit of a number in hexadecimal is simply computed by finding out . So what is ? Answer
, because for some . Example 3Find the smallest natural number that leaves a remainder of when divided by and a remainder of when divided by . Let our mystery number be . We require . Therefore, for some . Now consider . Therefore, we note that or in other words and for some . Let us choose to yield and therefore . The smallest number is indeed . We will examine more problems of this sort when we look at the Chinese remainder theorem. |