Beginning Activity Beginning Activity 1: Definition of Divides, Divisor, Multiple
In Section 1.2, we studied the concepts of even integers and odd integers. The definition of an even integer was a formalization of our concept of an even integer as being one that is “divisible by 2,” or a “multiple of 2.” We could also say that if “2 divides an integer,” then that integer is an even integer. We will now extend this idea to integers other than 2. Following is a formal definition of what it means to say that a nonzero integer \(m\) divides an integer \(n\text{.}\)
Definition.
A nonzero integer \(m\) divides an integer \(n\) provided that there is an integer \(q\) such that \(n = m \cdot q\text{.}\) We also say that \(m\) is a divisor of \(n\text{,}\) \(m\) is a factor of \(n\text{,}\) and \(n\) is a multiple of \(m\text{.}\) The integer 0 is not a divisor of any integer. If \(a\) and \(b\) are integers and \(a \ne 0\text{,}\) we frequently use the notation \(a \mid b\) as a shorthand for “\(a\) divides \(b\text{.}\)”
A Note about Notation.
Be careful with the notation \(a \mid b\text{.}\) This does not represent the rational number \(\dfrac{a}{b}\text{.}\) The notation \(a \mid b\) represents a relationship between the integers \(a\) and \(b\) and is simply a shorthand for “\(a\) divides \(b\text{.}\)”
A Note about Definitions.
Technically, a definition in mathematics should almost always be written using “if and only if.” It is not clear why, but the convention in mathematics is to replace the phrase “if and only if” with “if” or an equivalent. Perhaps this is a bit of laziness or the “if and only if” phrase can be a bit cumbersome. In this text, we will often use the phrase “provided that” instead.
The definition for “divides” can be written in symbolic form using appropriate quantifiers as follows: A nonzero integer \(m\) divides an integer \(n\) provided that \(\left( {\exists q \in \mathbb{Z}} \right)\left( {n = m \cdot q} \right)\text{.}\)
1.
Use the definition of divides to explain why 4 divides 32 and to explain why 8 divides \(-96\text{.}\)
2.
Give several examples of two integers where the first integer does not divide the second integer.
3.
According to the definition of “divides,” does the integer 10 divide the integer 0? That is, is 10 a divisor of 0? Explain.
4.
Use the definition of “divides” to complete the following sentence in symbolic form: “The nonzero integer \(m\) does not divide the integer \(n\) means that … .”
5.
Use the definition of “divides” to complete the following sentence without using the symbols for quantifiers: “The nonzero integer \(m\) does not divide the integer \(n \ldots \text{.}\)”
6.
Give three different examples of three integers where the first integer divides the second integer and the second integer divides the third integer.
As we have seen in Section 1.2, a definition is frequently used when constructing and writing mathematical proofs. Consider the following conjecture:
Let \(a\text{,}\) \(b\text{,}\) and \(c\) be integers with \(a \ne 0\) and \(b \ne 0\text{.}\) If \(a\) divides \(b\) and \(b\) divides \(c\text{,}\) then \(a\) divides \(c\text{.}\)
7.
Explain why the examples you generated in Exercise 6 provide evidence that this conjecture is true.
In Section 1.2, we also learned how to use a know-show table to help organize our thoughts when trying to construct a proof of a statement. If necessary, review the appropriate material in Section 1.2.
8.
State precisely what we would assume if we were trying to write a proof of the preceding conjecture.
9.
Use the definition of “divides” to make some conclusions based on your assumptions in Exercise 8.
10.
State precisely what we would be trying to prove if we were trying to write a proof of the conjecture.
11.
Use the definition of divides to write an answer to the question, “How can we prove what we stated inExercise 10?”