Chapter 4: Number Theory
4.1
Factors and
Divisibility
4.1.1. Connecting Factors and Multiples
4.1.1.1.
Definition of factor and multiple: If a and b are whole
numbers and ab = c, then a is a factor of c, b is a
factor of c, and c is a multiple of both a and b.
4.1.1.2.
Factors of 12: 1, 2, 3, 4, 6, 12 (finite set)
4.1.1.3.
Multiples of
4.1.1.4.
Finding
factors and multiples
4.1.1.4.1. Multiples on your calculator
4.1.1.4.2. Factors on your calculator
4.1.1.4.3.
Your turn p. 189: Do the practice and the reflect
4.1.1.4.4. Theorem:
Factor Test – To find all of the
factors of a number n, test only those natural numbers that are no greater than
the square root of the number, 
.
4.1.1.4.5.
Your turn p. 191: Do the practice and the reflect
4.1.2. Defining
Divisibility
4.1.2.1.
Definition of Divisibility: For whole numbers a and b,
a ¹ 0, a divides b, written
a|b, if and only if there is a whole number x so that
ax = b. Also, a is
a divisor of b or b is divisible by a.
Further, a 
b means that a does
not divide b.
4.1.3. Techniques
for Determining Divisibility
4.1.3.1.
Theorem: Divisibility of Sums – For natural numbers a, b, and c, if a|b and a|c, then a|(b + c)
4.1.3.2.
Theorem: Divisibility of 2, 5, and 10 –
4.1.3.2.1. A natural number n is divisible by 2 if and only if
its units digit is 0, 2, 4, 6, or 8.
4.1.3.2.2. A natural number n is divisible by 5 if and only if
its units digit is 0 or 5.
4.1.3.2.3. A natural number n is divisible by 10 if and only if
its units digit is 0.
4.1.3.3.
Definition of Even and Odd Numbers:
4.1.3.3.1. A whole number is even if and only if it is divisible
by 2.
4.1.3.3.2. A whole number is odd if and only if it is not
divisible by 2.
4.1.3.4.
Theorem: Divisibility Tests for 3 and 9 –
4.1.3.4.1. A natural number n is divisible by 3 if and only if
the sum of its digits is divisible by 3.
4.1.3.4.2. A natural number n is divisible by 9 if and only if
the sum of its digits is divisible by 9.
4.1.3.5.
Theorem: Divisibility of Products – For natural numbers a, b, and c, if a|c and b|c, and a and b have no common factors except 1, then ab|c.
4.1.3.6.
Theorem: A divisibility test for 6 – A natural number n is divisible by 6 if and only
if it is divisible by both 2 and 3.
4.1.3.7.
Your turn p. 195: Do the practice and the reflect
4.1.3.8.
Theorem: Divisibility tests for 4 and 8 –
4.1.3.8.1. A natural number n is divisible by 4 if and only if
the number represented by its last two digits is divisible by 4.
4.1.3.8.2. A natural number n is divisible by 8 if and only if
the number represented by its last three digits is divisible by 8.
4.1.3.9.
Theorem: Divisibility tests fro 7 and 11 –
4.1.3.9.1. A natural number n is divisible by 7 if and only if
the number formed by subtracting twice the last digit from the number formed by
all digits but the last is divisible by 7.
4.1.3.9.2. A natural number n is divisible by 11 if and only if
the sum of the digits in the even-powered places minus the sum of the digits in
the odd-powered places is divisible by 11.
4.1.3.10.
Your turn p. 197: Do the practice and the reflect
4.1.3.11. Theorems:
Divisibility – for natural numbers
a, b, and c,
4.1.3.11.1.
If a|b and a|c, then a|(b-c)
4.1.3.11.2.
If a|b and c is any natural number, then a|bc
4.1.3.11.3.
If a(b + c), and
a|b, then a|c
4.1.3.11.4.
If a|(b – c),
and a|b, then a|c
4.1.4. Using
Factors to Classify Natural Numbers
4.1.4.1.
even numbers
4.1.4.2.
odd numbers
4.1.4.3.
squares
4.1.4.4.
perfect number –
sum of the factors less than the number equals the number: 1 + 2 + 3 = 6, then
6 is a perfect number
4.1.4.5.
deficient number
– sum of the proper factors is less than the number
4.1.4.6.
abundant number
– sum of the proper factors is greater than the number
4.1.4.7.
amicable – sum
of the proper factors of the first number equals the second number and if the
sum of the proper factors of the second number equals the first number
4.1.4.8.
Your turn p. 200: Do the practice and the reflect