Divisibility

    The concept of divisibility is the main difference between the integers and the rational numbers. Indeed, the sum, difference, and product of two integers is an integer, but the quotient of two integers may not be an integer.

Definition (Divisibility) If and are integers with we say that divides written when there exists an integer such that If divides then we also say that is a divisor of is a factor of and that is a multiple of

Example (Divisibility) Here are some examples of divisibility: (a) since (b) since (c) since (d) since and (e) since

    There are many elementary properties of divisibility that follow from the definition of divisibility. Here are some of them with proof.

Proposition (Properties of Divisibility) Assume that and are integers.

    (i) If and then
    
    (ii) If and   then for any integers and
    
    (iii) If and then
    
    (iv) If and then
    
    (v) If then for any positive integer
    
    Proof. For part (i), suppose there exists and such that and Then and so
    For part (ii), suppose there exists integers and such that and Then for any integers and we have and so
    For part (iii), suppose there exists integers and such that and Then we have Thus, and so Whence,
    For part (iv), suppose there exists integers and such that and Then, since and so
    For part (v), we will use mathematical induction. Since certainly implies the case for is trivial. Assume that holds, then there exists an integer such that Then where and are some integers. Whence,   as desired.  

• print this page
• pages linking here
• related pages