What is a factor of a number?
What numbers have a factor of 1?
A factor, or divisor, is a number that divides a larger number without a remainder.
All integers have 1 as a factor.
*** Factors can be negative.
Examples:
7 is a factor of 56 because 56/7 = 8
7 is a factor of 42 because 42/7 = 6
What is a multiple?
A multiple is the product of any number and an integer. That integer divides the multiple evenly, without a remainder.
*** Multiples can be negative.
Examples:
Multiples of 7 include 21, 7, 7, 21, 28, 35, etc.
Greatest Common Factor
The Greatest Common Factor (GCF) of two numbers is the largest factor the two numbers have in common.
How do you find the GCF of two numbers?

List all factors of each given number separately

Find the largest factor that appears in all lists
Example:
Find the GCF of 6 and 9.
The factors of 6 are 1, 2, 3, 6; the factors of 9 are 1, 3, 9. The GCF is 3.
What is prime factorization?
Prime factorization is a way to present a positive integer as a product of prime numbers.
Example:
Factor 96 into prime factors.
96 = 2 x 2 x 2 x 2 x 2 x 3
Write the product in exponential form.
96 = 2^{5} x 3
How do you find the GCF of two numbers using prime factorization?

Factor each number into primes

Select common factors by pairs

Multiply these factors

The result is the GCF
Example:
Find the GCF of 24 and 36.
24 = 2 x 2 x 2 x 3
36 = 2 x 2 x 3 x 3
The GCF is 2 x 2 x 3 = 12
Least Common Multiple
The Least Common Multiple (LCM) is the smallest multiple two numbers have in common.
How do you find the LCM using successive multiplication?

Multiply the bigger number by 1, 2, 3, ...

Repeat with the smaller number

Check both lists to see the smallest common multiple
Example:
Find the LCM of 10 and 15.
Multiples of 15 are 15, 30, 60, 90, etc.
Multiples of 10 are 10, 20, 30, 40, etc.
30 is the LCM of 10 and 15.
How can you find the LCM of two numbers by using the GCF?

Find the product of the two numbers

Divide the product by the GCF of both numbers
Example:
Find the LCM of 10 and 15.
First step... 10 x 15 = 150.
The GCF of 10 and 15 is 5.
150 ÷ 5 = 30 is the LCM of 10 and 15.
How can you find the LCM of two numbers using prime factorization?

Factor each number into primes

Take the number with most factors to start a list

Add to the list the missing factors from other numbers
Example:
Find the LCM of 6, 8 and 18.
6 = 2 x 3 ; 8 = 2 x 2 x 2 ; 18 = 2 x 3 x 3
Let's start the list with 8: 2 x 2 x 2. You don't need to use 2 from other numbers. Add 3 x 3 from 18 to the list.
The LCM = 2 x 2 x 2 x 3 x 3 = 72
How do you find the LCM of 9 and 12 using prime factors with larger exponents?
Factor both numbers into primes:
9 = 3 x 3 ; 12 = 2 x 2 x 3
Write each product using exponents:
3 x 3 = 3^{2} ; 2 x 2 x 3 = 2^{2} x 3
Multiply factors with the larger exponent:
3^{2} x 2^{2} = 36
The LCM of 9 and 12 is 36
set
A set is a collection of numbers.
*** The distinct objects within a set can be called elements.
{ } are used to denote a set.
What is the difference between a finite set and an infinite set?
If you can list all the elements of a set, it is finite.
Example:
{4,10,16, 20} is a finite set of 4 numbers.
Otherwise, the set is infinite.
Example:
{1, 2, 3, 4...} is the set of all natural numbers and is infinite.
empty set
A set that has no elements is an empty set.
union of sets
The union of sets is the set of all the elements from those sets, without repetition.
*** U denotes union.
Example:
Set X contains all prime numbers less than 10. Set Y contains all odd numbers less than 10.
X = {2, 3, 5, 7} ; Y = {1, 3, 5, 7, 9}
X U Y = {1, 2, 3, 5, 7, 9}
intersection of sets
An intersection of sets is the set of elements common to all sets.
*** An element must be in all sets to be in the intersection.
∩ is used to denote intersection.
Example:
Set X is all prime numbers less than 10. Set Y is all odd numbers less than 10. Find the intersection set.
X = {2, 3, 5, 7} ; Y = {1, 3, 5, 7, 9}
X ∩ Y = {3, 5, 7}
What is a Venn Diagram?
A Venn Diagram is made up of two or more overlapping circles. It is used to show relationships between sets. The overlapping area (intersection) shows common elements between sets.
How would you solve this problem using a Venn diagram?
There are 35 students in a class. 12 are taking French. 20 are taking Spanish. If 3 students are taking both French and Spanish, how many students don't take any language classes?
We can draw a Venn Diagram and label the information.
 "C"  students taking both French and Spanish = 3
 "A"  students who take only French = 12  3
 "B"  students who take only Spanish = 20  3
A + C + B = 3 + 9 + 17 = 29 (these are students who take either French or Spanish or both)
That leaves 6 students unaccounted for. These are the ones who do not take any language classes.
sequence
A sequence, or progression, is a list of numbers in a specified pattern.
The following are types of sequences you might see on the SAT: arithmetic, geometric, and Fibonacci sequences.
arithmetic sequence
In an arithmetic sequence, the difference between any two consecutive terms is constant.
This constant is called the "common difference" d.
Examples:
3, 6, 9, 12, 15... (d = 3)
10, 6, 2, 2, 6... (d =  4)
How do you find the common difference (d) in an arithmetic sequence?
The common difference (d) can be calculated by finding the difference of any two consecutive terms in an arithmetic sequence.
d = a_{n}  a_{n1}
Example:
In 2, 6, 10, 14, 18.... sequence common difference (d) equals 4.
Let's find missing terms in the sequence below:
3, 5, 7, ..., 11, ...
This is an arithmetic sequence because you add a constant to get from one term to the next. The constant (or common difference d) is 2 (4  2; 6  4).
1st term: 3
2nd term: 5 = 3 + 2
3rd term: 7 = 5 + 2 = 3 + 2 + 2 = 3 + (2 * 2)
4th term: 9 = 7 + 2 = 3 + 2 + 2 + 2 = 3 + (2 * 3)
5th term: 11 = 9 + 2 = 3 + 2 + 2 + 2 + 2 = 3 + (2 * 4)
6th term: 13 = 11 + 2 = 3 + (2 * 5)
7th term: 15 = 13 + 2 = 3 + (2 * 6)
Let's examine how we found the 6th term. 3 is the first term. 2 is the common difference. 5 is the number of terms minus 1. So, the formula to find the 6th term is: 1st term + d * (# of terms  1).
What is the formula for finding the nth term of an arithmetic sequence?
a_{n} = a_{1} + d * (n  1)
 a_{1} is the first term of the sequence
 d is the common difference
 n is the number of terms in the sequence
Example:
Find the 4th term of an arithmetic sequence with the first term 2 and the common sum d = 3.
a_{4} = 2 + 3 * (4  1) = 2 + 9 = 11
How do you find the sum of the terms of an arithmetic sequence (also called the sum of a series)?
To find the sum of a series, use the following formula:
S_{n} = n * ^{1}/_{2}(a_{1} + a_{n})
 S_{n} is the sum of the first n numbers
 a_{1 }is the first term
 a_{n} is the last term
 n is the number of terms
geometric sequence
In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed value of "common ratio" r.
*** The common ratio (r) cannot be zero.
Examples:
3, 6, 12, 24, 48... (r = 2)
16, 8, 4, 2, 1... (r = ^{1}/_{2})
How do you find the common ratio (r) in a geometric sequence?
The common ratio (r) can be calculated by dividing any two consecutive terms in a geometric sequence.
Let's find the missing terms in the sequence below:
1, 2, 4, 8, ..., 32, ...
This is a geometric sequence because there is a constant ratio between any two consecutive terms. The ratio is 2 (8 ÷ 4).
1st term: 1
2nd term: 2 = 1 * 2
3rd term: 4 = 2 * 2 = 1 * 2^{2}
4th term: 8 = 4 * 2 = 2 * 2 * 2 = 1 * 2^{3}
5th term: 16 = 8 * 2 = 2 * 2 * 2 * 2 = 1 * 2^{4}
6th term: 32 = 16 * 2 = 2 * 2 * 2 * 2 * 2 = 1 * 2^{5}
7th term: 62 = 32 * 2 = 2 * 2 * 2 * 2 * 2 * 2 = 1 * 2^{6}
Let's examine how we found the 7th term. 1 is the first term. 2 is the common ratio. Power of 6 indicates the number of terms minus 1.
What is the formula for finding the nth term of a geometric sequence?
a_{n} = a_{1} * r^{ (n 1)}
 a_{1 }is the first term of the sequence
 n is the number of terms in the sequence
 r is the common ratio
Example:
Find the 5th term of a geometric sequence with the common ratio 2 and the first term 3.
a_{5} = 3 x 2^{(51)} = 3 x 2^{4} = 48
Fibonacci sequence
In the Fibonacci sequence, each successive number is the sum of the previous two.
The first two numbers in the sequence are 0 and 1.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
What is the formula for finding the nth term of the Fibonacci sequence?
a_{n} = a_{n1} + a_{n2}