# Chapter 4

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McGill University

Mathematics & Statistics (Sci)

MATH 111

Nabil Kahouadji

Winter

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MATH 111 – Winter 2012
4: INTEGER AND NUMBER THEORY
4.1 Integers and Operations of Addition and Subtraction
4.1.1 Representations of Integers
- negative integers are opposites of positive integers, & vice versa
o ie/ the opposite of 2 is -2, and the opposite -3 is 3
4.1.2 Integer Addition
4.1.3 Number-Line Model
4.1.4 Charged-Field Model for Addition
- a field has 0 charges if it has the same number of positive & negative charges
o ie/ 5 + (-5) = 0
4.1.5 Absolute Value
absolute value: the distance between the point corresponding to an integer and 0; the absolute value of
an integer x, written |x| is equal to x if x ≥ 0, and is equal to –x if x < 0
4.1.6 Properties of Integer Addition
closure property of addition of integers: given integers a and b, a + b is a unique integer MATH 111 – Winter 2012
commutative property: given the integers a and b, a + b = b + a
associate property: given the integers a, b and c, (a + b) + c = a + (b + c)
additive identity element: 0 is the unique integer such that, for all integer a, 0 + a = a = a + 0
uniqueness of the additive inverse: for every integer a, there is a unique –a, the additive inverse of a,
such that a + (-a) = 0 = -a + a
additive inverse properties: for any integers a and b: -(-a) = a; -a + (-b) = -(a + b)
4.1.7 Integer Subtraction
subtraction: for integers a and b, a – b is the unique integer n such that a = b + n
- for all integers a and b, a – b = a + (-b)
4.2 Multiplication and Division of Integers
multiplication for any whole number: (-a)(-b) = ab; (-a)b = b(-a) = -(ab)
closure property of integer multiplication: ab is a unique integer
commutative property: ab = ba
associate property: (ab)c = a(bc)
multiplicative identity element: 1 is the unique integer such that for all integers a, 1 x a = a = a x 1
distributive properties of multiplication over addition for integers: a(b + c) = ab + bc
zero multiplication property: 0 is the unique integer such that for all integers a, a x 0 = 0 = 0 x a
- for every integer a, (-1)a = -a
- for all integers a and b, (-a)b = -(ab) and (-a)(-b) = ab
distributive property of multiplication over subtraction of integers: for any integers a, b and c, a(b – c)
= ab – ac and (b – c)a = ba – ca
4.2.2 Integer Division
integer division: if a and b are any integers a ÷ b is the unique integer c, if it exists, such that a = bc
4.2.3 Order of Operations on Integers
- when addition, subtraction, multiplication, division and exponentiation appear without parentheses,
exponentiation is done first in order from right to left, then multiplications and divisions in the order of MATH 111 – Winter 2012
their appearance from left to right and then additions and subtractions in the order of their appearance
from left to right
- arithmetic operations that appear inside parentheses must be done first
4.2.4 Ordering Integers
less than for integers: for any integers a and b, a is less than b, written a < b, if, and only if, there exists a
positive integer k such that a + k = b
- a < b (or equivalently, b > a) if, and only if, b – a is equal to positive integer, that is b – a is greater
than 0
- if x < y and n is an integer, then x + n < y + n
- if x < y, then –x > –y
- if x < y and n > 0, then nx < ny
- if x < y and n < 0, then nx > ny
4.3 Divisibility
divides: if a and b are any integers, then b divides a, written b|a, if, and only if, there is a unique integer
q such that a = bq
- can also say: b is divisor/ factor of a, a is a factor/divisible of b
- b does not divide a denoted b | a
- for any integers a and b, if d|a and n is any integer, then d|na
- for any integers a, b and d with d ≠ 0
o if d|a and d|b then d|(a + b)
o if d|a and d|b, then d|(a + b)
o if d|a and d|b, then d|(a – b)
o if d|a and d|b, then d|(a – b)
4.3.1 Divisibility Rules
divisibility test for 2: an integer is divisible by 2, if, and only if, its units digit is divisible by 2; that is, if,
and only if, the units digits is 0, 2, 4, 6 or 8
divisibility test for 5: an integer is divisible by 5, if, and only if, its units digit is divisible by 5; that is if,
and only if, the units digits is 0 or 5
divisibility test for 10: an integer is divisible by 10, if and only if, the units digit is divisible by 10; that is,
if and only if, the units digit is 0
divisibility test for 4: an integer is divisible by 4, if and only if, the last

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