CSCI 124 and CSCI 224/Vora/GWU 1
CSCI 124/224: Discrete Structures II: Divisibility
In this section, we study the divisibility of one integer by another. You should already be familiar with the
basic ideas, but need to be able to prove them, using simple logical steps. Again, there is no way to become
familiar with proofs without practice. Make sure you do the problems from the handout.
∃denotes “there exists”.
Definition For integers aand b,a6= 0,ais said to divide bif ∃integer msuch that b=ma. This is denoted
as a|b.bis said to be divisible by a. Also, ais a factor or divisor of b, which is a multiple of a.
Examples: 2|1024,3|171,5-1024 (5does not divide 1024).
Theorem Divisibility is transitive. That is, for integers a, b, c such that a6= 0 and b6= 0, if a|band b|c, then
a|c.
Proof: Suppose a6= 0 and b6= 0. Suppose further that a|band b|c.
a|b, b|c
⇒ ∃ m1, m2, s.t.b =m1a, c =m2b
⇒c=m1m2a
⇒a|c
Example: 9|126 and 126|378, hence 9|378.
Division Theorem If aand bare integers such that b > 0,∃unique integers q(the quotient) and r(the
remainder) such that a=bq +r, with 0≤r < b.
We do not study its proof.
Example: a= 7,b= 3,r= 1,q= 2. Another example: a=−9,b= 5,q=−2,r= 1 (and not q=−1
and r=−4. Why not?)
An even integer is an integer that is divisible by 2. That is, there is an integer msuch that the even integer
may be written as 2m.
An odd integer is one that is not divisible by 2. Its remainder is 1when divided by 2, hence ∃unique integer
qsuch that the odd integer may be written as 2q+ 1.
If the product of two integers is the integer 1, both integers are either 1or −1.
