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 12: 12, 24, 36, 48, … (infinite set)
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. 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, n .
4.1.2. Defining Divisibility
4.1.2.1. Definition of Divisibility: For whole numbers a and b, a ≠ 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. Theorem: Divisibility tests for 4 and 8 –
4.1.3.7.1. A natural number n is divisible by 4 if and only if the number
represented by its last two digits is divisible by 4.
4.1.3.7.2. A natural number n is divisible by 8 if and only if the number
represented by its last three digits is divisible by 8.
4.1.3.8. Theorem: Divisibility tests for 7 and 11 –
4.1.3.8.1. 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.
4.1.3.8.2. 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.
4.1.3.9. Theorems: Divisibility – for natural numbers a, b, and c,
4.1.3.9.1. If a|b and a|c, then a|(b-c)
4.1.3.9.2. If a|b and c is any natural number, then a|bc
4.1.3.9.3. If a(b + c), and a|b, then a|c
4.1.3.9.4. 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
4.1.5. Problems and Exercises p. 217
4.1.5.1. Home work: 2-4, 6, 8-15, 17-20, 23, 29, 33, 34, 35, 38