Lesson 1: Factors and Multiples of Whole Numbers
Lesson 2: Perfect Squares and Cubes, Square and Cube Roots
Lesson 3: Using Factors and Multiples With Fractions
Key words: prime factors, greatest common factor (GCF), least common multiple
(LCM), square root, cube root
Real Numbers(ℝ)
All of the above number sets
Irrational Numbers (Q’) Any non-terminating, non-repeating decimal number
Rational Numbers (Q)
Any number that can be written as a ratio (fraction)
Examples for you to do:
Find out the irrational numbers among following:
1, 0, 1.365756, 3/5, 0.1235813…,
Determine the number sets of each number showing below:
a) 0
b) 2.7
c) 5/4
d) 7
e) 7.194757142857…
Factors of a number are numbers that multiply together to make that number.
For example:
6 and 4 are factors of 24 (6 x 4 = 24).
24 has the following factors: 1, 2, 3, 4, 6, 8, 12, 24
Prime Numbers and Factors
Any whole number greater than one that has only two distinct factors (one, and itself) is
called a prime number
Example: 13 is prime because it’s only factors are 1 and 13.
Numbers greater than 1 that are not prime are called composite numbers.
When the factors of a number are also prime, they are called prime factors.
Example: 12 has prime factors 2 x 2 x 3. We call this the prime factorization of 12.
*
The number
1 is not a prime number because it is not divisible by
any whole numbers other than itself
The number
0
is not prime because it does not have 2 distinct
factors
Greatest Common Factor (GCF)
The greatest common factor of two or more whole numbers is the largest whole
number that is a factor of two or more numbers.
Example: The GCF of 12 and 18 is 6.
12 has factors 1, 2, 3, 4, 6, 12
18 has factors 1, 2, 3, 6, 9, 18
The largest shared factor between these two number is 6, so 6 is the GCF. Here are
some techniques to finding the GCF of different numbers.
Least Common Multiple (LCM)
The least common multiple of two or more whole numbers is the smallest whole
number that is a multiple of two or more whole numbers.
For example, the least common multiple of 12 and 18 is 36 because it is the
smallest whole number that is a multiple of both 12 and 18.
Examples for us to do:
Determine the GCF of 15 and 9, use 3 different ways.
Determine the GCF of 12, 8, and 36.
Determine the GCF of 12, 8, and 9.
Square Roots and Cube Roots
The square root of a number, x, is the number, s, such that s x s = x.
For example, the square root of 4, is 2 (2 x 2 = 4)
The cube root of a number, x, is the number q, such that q x q x q = x.
For example, the cube root of 8 is 2 (2 x 2 x 2 = 8)
You should be able to recognize perfect squares and cubes that are 100
or less, and, consequently, know their square and cube roots.
For example, you should be able to recognize 49 as a perfect square (49
= 7 x 7) and 27 as a perfect cube (3 x 3 x 3 = 27).
Knowing these allows you to state that the positive square root of 49 is
7, and the cube root of 27 is 3.
The symbol that we use to show the operation of taking the positive
square root is √, and the symbol for cube root is 3√. These symbols
are called radical signs.
The symbol √ represents the positive square root so that √49 = 7.
The negative square root is represented by -√ so that, for example, √49 = -(7) = -7.
The symbol for both the positive and negative square roots is ±√ so
that, for example, ±√49 = ±7.
The square root of a negative number is not a real number so that, for
example, √-49 does not exist as a real number. There are however
cube roots of positive and negative numbers so that , for example, 3√
27 = 3 and 3√-27 = -3.
Examples for us to do:
Determine the cube root of 343
Determine the negative square root of 3375
Determine the cube root of 2744