1. Divisibility in the integers
The symbol Z is used to denote the set of integers. This originates from the German word
for number, which is Zahl. The set of integers is the set of all whole numbers, their negatives,
and 0. In set notation, we write
Z = {. . . , −2, −1, 0, 1, 2, . . . }.
Other number systems include the natural numbers N (these are the positive integers), N0
(the natural numbers and zero), the rational numbers Q (all fractions of integers), the real
numbers R, and the complex numbers C. In this course, we are primarily interested in Z,
and occasionally Q.
We sometimes write a ∈ Z instead of saying “a is an integer.” Other notation we will
sometimes use: ∀ means “for all” and ∃ means “there exists”.
We recalled the commutative, associative, and distributive laws for integer arithmetic. I.e.,
the commutative law of addition says that x + y = y + x for all integers x, y. Multiplication
is also commutative. The associative law of multiplication says that x(yz) = (xy)z for
all integers x, y, z. Similarly, addition is associative. And the distributive law says that
x(y + z) = xy + xz for all x, y, z ∈ Z.
What does it mean for one integer to divide another (“evenly”)? Tiffany came up with
the following definition:
Definition: Let a, b ∈ Z. We say that a divides b if the fraction ab is an integer. Alternatively, we can say that a divides b if there exists some integer c such that b = ac.
Notation: We often will write a| b for “a divides b.”
Example: 2| 4 (i.e., 2 divides 6) since 6 = 2(3). (Using the notation in the definition,
a = 2, b = 6 and c = 3.)
Example: 1 | 6 since 6 = 1(6). (Here, a = 1, b = 6, and c = 6.)
We then bravely proved our first “theorem” (if we dare call it that!):
Theorem: Let n be any integer. Then 1| n and −1| n.
Proof: (class) This is clear, since for any integer n we have n = (1)(n) and n = (−1)(−n).
Homework: Let a, b, and c be integers.
(1) If a| b, does a| bc?
(2) If a| bc, must a| b or a| c?
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