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NRF 10 -
3.1 Factors and Multiples of Whole Numbers
Where does our number system come from?
The simple answer is from the Hindu-Arabic system (a base 10 place holder system) which
was influenced by the fact that we have 10 fingers!
Our Number System has become more clearly defined with history. We now categorize
numbers as follows:


Note: The definition of a Rational Number is: Q  ab a, b I , b  0
Translation: If it can be made into a fraction it is Rational!
Classify each of the following numbers:
1. ‘5’
Natural, Whole, Integer, Rational, Irrational, Real
2. ‘1.6’
Natural, Whole, Integer, Rational, Irrational, Real
3. ‘ 8 ’
Natural, Whole, Integer, Rational, Irrational, Real
4. ‘ 2.3
Natural, Whole, Integer, Rational, Irrational, Real
5. ‘-4’
Natural, Whole, Integer, Rational, Irrational, Real
6. ‘-1.25’
Natural, Whole, Integer, Rational,
Irrational, Real
7. ‘  ’
8. ‘0.121212 . . .’
9. ‘
’
10. ‘
’
Natural,
Natural,
Natural,
Natural,
Irrational,
Irrational,
Irrational,
Irrational,
Whole,
Whole,
Whole,
Whole,
Integer,
Integer,
Integer,
Integer,
Rational,
Rational,
Rational,
Rational,
Real
Real
Real
Real
Prime Numbers
When a number has exactly two factors, 1 and itself, the number is a prime number.
Examples: 2, 3, 5, ___, ___, ___, ___, ….
Composite Numbers have more than two factors
Note: - The number 1 is a special number – neither prime or composite
- The number 0 is composite – infinite number of factors
Prime Factorization
Every Natural number can be re-written as the product of prime numbers. This is the prime
factorization of a number. Factor Trees can be used to break a number down to its prime
factorization
Examples:
12
75
1750
22302
Divisibility Tests/Rules are great at helping break up larger Natural numbers into their
prime factors:
Recall:
Divisibility Rule for 2: ____________________________________
Divisibility Rule for 3: ____________________________________
Divisibility Rule for 4: ____________________________________
Divisibility Rule for 5: ____________________________________
Divisibility Rule for 6: ____________________________________
Divisibility Rule for 8: ____________________________________
Divisibility Rule for 9: ___________________________________
Divisibility Rule for 10: __________________________________
The Greatest Common Factor (GCF)
The GCF of two or more numbers is the greatest factor the numbers have in common.
Example 1: What is the GCF of 76 and 148?
Solution:
Method 1: Finding the GCF by listing the factors of each number
Step1 : List the factors of each number
76 – 1, 2, 4, 19, 38, 76
148 - 1, 2, 4, 37, 74, 148
Step 2: Identify the greatest of the common factors
GCF(76,148) = 4
4 is the “largest number that divides both numbers”
Method 2: Finding the GCF using the prime factorizations of each numbers
Step 1 : Break each number into the product of its primes (prime factorization)
76 = 2 x 38 = 2 x 2 x 19
148 = 2 x 74 = 2 x 2 x 37
Step 2 : Multiply the common factors of each prime factorization list
Since both have two 2’s in common, the GCF(76,148) = 2 x 2 = 4
Example 2: What is the GCF of 70 and 84?
Example 3: What is the GCF of 420, 555, and 615?
Least Common Multiple (LCM)
The LCM of two or more numbers is the least (smallest) number that is divisible by each
number.
Example: What is the LCM of 32 and 50?
Method 1 – Using the list of multiples of each number
Multiples of 32 – 32, 64, 96, 128, 160, 192, 224, 256, 288, 320, 352, 384, 416, 448, 480,
512, 544, 576, 608, 640, 672, 704, 736, 768, 800, 832, 864,…
Multiples of 50 – 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600, 650, 700, 750,
800, 850, 900,…
The LCM(32,50) is 800.
A lot of work….is there a more efficient way of finding the answer?
Yes, there is! Consider the next method:
Method 2 – Finding the LCM using the prime factorizations of each number
32 = 2 x 2 x 2 x 2 x 2 = 25
50 = 2 x 5 x 5 = 2 x 52
LCM(32,50) = 25 x 52 = 32 x 25 = 800
(The LCM is the product of the highest power of all unique primes)
Example 2: Find the least common multiple of 20, 36, 38.
20 = 2 x 2 x 5
36 = 2 x 2 x 3 x 3
114 = 2 x 3 x 19
= 22 x 5
= 22 x 32
= 2 x 3 x 19
LCM = 22 x 32 x 5 x 19 = 3420
Example 3: Find the least common multiple of 28, 40, 44.
Problems:
1. A builder wants to cover a wall measuring 9 ft. by 15 ft. with square
pieces of plywood.
a) What is the side length of the largest square that could be used to
cover the wall? Assume the squares cannot be cut.
b) How many square pieces of plywood would be needed?
2. Bill and Betty do chores at home. Bill mows the lawn every 8 days,
and Betty bathes the dog every 14 days. Suppose Bill and Betty do
their chores today. How many days will pass before they both do their
chores on the same day again?