Factors, Multiples, Sets, and Sequences

Question 1

What is a factor of a number?

What numbers have a factor of 1?

Answer 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

Question 2

What is a multiple?

Answer 2

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.

Question 3

Define:

Greatest Common Factor

Answer 3

The Greatest Common Factor (GCF) of two numbers is the largest factor the two numbers have in common.

Question 4

How do you find the GCF of two numbers?

Answer 4

• 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.

Question 5

What is prime factorization?

Answer 5

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 = 2⁵ x 3

Question 6

How do you find the GCF of two numbers using prime factorization?

Answer 6

• 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

Question 7

Define:

Least Common Multiple

Answer 7

The Least Common Multiple (LCM) is the smallest multiple two numbers have in common.

Question 8

How do you find the LCM using successive multiplication?

Answer 8

• 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.

Question 9

How can you find the LCM of two numbers by using the GCF?

Answer 9

• 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.

Question 10

How can you find the LCM of two numbers using prime factorization?

Answer 10

• 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

Question 11

How do you find the LCM of 9 and 12 using prime factors with larger exponents?

Answer 11

Factor both numbers into primes:

9 = 3 x 3 ; 12 = 2 x 2 x 3

Write each product using exponents:

3 x 3 = 3² ; 2 x 2 x 3 = 2² x 3

Multiply factors with the larger exponent:

3² x 2² = 36

The LCM of 9 and 12 is 36

Question 12

Define:

set

Answer 12

A set is a collection of numbers.

*** The distinct objects within a set can be called elements.

{ } are used to denote a set.

Question 13

What is the difference between a finite set and an infinite set?

Answer 13

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.

Question 14

Define:

empty set

Answer 14

A set that has no elements is an empty set.

Question 15

Define:

union of sets

Answer 15

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}

Question 16

Define:

intersection of sets

Answer 16

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}

Question 17

What is a Venn Diagram?

Answer 17

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.

Question 18

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?

Answer 18

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.

Question 19

Define:

sequence

Answer 19

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.

Question 20

Define:

arithmetic sequence

Answer 20

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)

Question 21

How do you find the common difference (d) in an arithmetic sequence?

Answer 21

The common difference (d) can be calculated by finding the difference of any two consecutive terms in an arithmetic sequence.

d = a_n - a_n-1

Example:

In 2, 6, 10, 14, 18…. sequence common difference (d) equals 4.

Question 22

Let’s find missing terms in the sequence below:

3, 5, 7, …, 11, …

Answer 22

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).

Question 23

What is the formula for finding the nth term of an arithmetic sequence?

Answer 23

a_n = a₁ + d * (n - 1)

• a₁ 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.

a₄ = 2 + 3 * (4 - 1) = 2 + 9 = 11

Question 24

How do you find the sum of the terms of an arithmetic sequence (also called the sum of a series)?

Answer 24

To find the sum of a series, use the following formula:

S_n = n * ¹/₂(a₁ + a_n)

• S_n is the sum of the first n numbers
• a₁ is the first term
• a_n is the last term
• n is the number of terms

Question 25

Define:

geometric sequence

Answer 25

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)

Question 26

How do you find the common ratio (r) in a geometric sequence?

Answer 26

The common ratio (r) can be calculated by dividing any two consecutive terms in a geometric sequence.

Question 27

Let’s find the missing terms in the sequence below:

1, 2, 4, 8, …, 32, …

Answer 27

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 * 2²
• 4th* term: 8 = 4 * 2 = 2 * 2 * 2 = 1 * 2³
• 5th* term: 16 = 8 * 2 = 2 * 2 * 2 * 2 = 1 * 2⁴
• 6th* term: 32 = 16 * 2 = 2 * 2 * 2 * 2 * 2 = 1 * 2⁵
• 7th* term: 62 = 32 * 2 = 2 * 2 * 2 * 2 * 2 * 2 = 1 * 2⁶

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.

Question 28

What is the formula for finding the nth term of a geometric sequence?

Answer 28

a_n = a₁ * r ^{(n -1)}

• a_<em>1</em> 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.

a₅ = 3 x 2^(5-1) = 3 x 2⁴ = 48

Question 29

Define:

Fibonacci sequence

Answer 29

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, …

Question 30

What is the formula for finding the nth term of the Fibonacci sequence?

Answer 30

a_n = a_n-1 + a_n-2

Question 31

List all factors of 12.

Find their sum.

Answer 31

Factors of 12 are

{1, 2, 3, 4, 6, 12}

1 + 2 + 3 + 4 + 6 + 12 = 28

The sum of all factors of 12 is 28.

Question 32

The sum of three consecutive integers is 9.

How many distinct prime factors are there in the product of these three integers?

Answer 32

The product has two distinct prime factors.

The only three consecutive numbers that add up to 9 are 2, 3, and 4. The product of these numbers is equal to 24.

The prime factors of 24 are 2, 2, 2, and 3.

Of these, only two are distinct: 2 and 3.

Question 33

Find the GCF of 28 and 52.

*** Use the method of listing all factors.

Answer 33

The GCF is 4.

Factors of 28: {1, 2, 4, 7, 14, 28}

Factors of 52: {1, 2, 4, 13, 26, 52}

Question 34

Use prime factorization to find the GCF of 20 and 44.

Answer 34

The GCF of 20 and 44 is 4.

Prime factors of 20 are 2, 2, and 5. Prime factors of 44 are 2, 2, and 11.

Select common factors in pairs and multiply them:

GCF = 2 x 2 = 4

Question 35

You and your friend are running laps on a track. You complete a lap every 40 seconds. Your friend completes a lap every 48 seconds. How many seconds will it take both of you to be at the starting line again if you start together?

Answer 35

In 240 second, both of you will meet at the starting line again.

To solve this problem, find the LCM of 40 and 48. Chose any of the 3 methods reviewed in this deck. We will use prime factorization here.

40 = 2³ x 5 ; 48 = 2⁴ x 3

The LCM is a product of 2⁴, 3 and 5.

The LCM = 240.

Question 36

What is the difference between the union of sets and the intersection of sets?

Answer 36

The union set includes all the elements that are in either or all sets.

The intersection is the set of common elements to all sets.

Question 37

Set X is all prime factors of 33; Set Y is all prime factors of 56.

What is the union of these sets?

X U Y = ?

Answer 37

The union of sets X and Y is:

X U Y = {2, 3, 7, 11}

Set X = {3, 11}… all prime factors of 33

Set Y = {2, 2, 2, 7}… all prime factors of 56

Question 38

How many different elements are in the union of the following sets?

Set A: all factors of 12.

Set B: all prime factors of 12.

Answer 38

The union of sets A and B contains 6 different elements.

The factors of 12 are {1, 2, 3, 4, 6, 12}.

The prime factors of 12 are {2, 2, 3}.

A U B = {1, 2, 3, 4, 6, 12} which contains 6 elements.

*** The union of two sets is the set of all the elements from both sets, without repetition.

Question 39

What is the intersection of sets A and B?

Set A: all factors of 20.

Set B: the first five multiples of 2.

Answer 39

The intersection of sets A and B is {2, 4, 10}.

• Set A* = {1, 2, 4, 5, 10, 20} … factors of 20.
• Set B* = {2, 4, 6, 8, 10} … first five multiples of 2.

*** The intersection of two sets is a set containing the elements common to both sets.

Question 40

What is the intersection of the set of all positive integers and the set of all even numbers?

Answer 40

The intersection of the set of all positive integers and the set of all even numbers is the set of all even positive integers.

Question 41

Find the intersection of the two sets below:

• Set A consists of all factors of 12
• Set B consists of first four multiples of 3

Answer 41

The intersection of sets A and B is:

A ∩ B = {3, 6, 12}

Factors of 12:

{1, 2, 3, 4, 6, 12}

First four multiples of 3:

{3, 6, 9, 12}

Question 42

Of the sequences below, which one is arithmetic?

(a) 2, 6, 12, 48
(b) -12, -9, -7, -5
(c) 5, 10, 20, 40
(d) 3, 6, 9, 12, 15

Answer 42

(d) 3, 6, 9, 12, 15

is an arithmetic sequence because there is a common difference (d = 3) between each of its consecutive terms.

Question 43

The nth term of a certain arithmetic sequence is defined to be 2n+ 8.

What is the difference between the 47th and the 35th term of this sequence?

Answer 43

The difference is 24.

Subsitute n into the formula with the number of terms in question.

• 47*th term = 2 x 47 + 8
• 35*th term = 2 x 35 + 8

You don’t have to calculate the expressions to see that the difference is 2 x 12, or 24.

Question 44

Of the four sequences below, one is geometric. Which is it?

(a) 2, 6, 12, 48
(b) -12, -9, -7, -5
(c) 5, 10, 20, 40
(d) 3, 6, 9, 12, 15

Answer 44

(c) 5, 10, 20, 40

is a geometric sequence because there is a common ratio (r = 2) between its consecutive terms.

Question 45

What is the 6th term in the following geometric sequence?

2, 4, 8, 16, …

Answer 45

The 6th term in this sequence is 64.

First, find the common ratio (r): 4/2 = 2.

Then, use the nth term formula for geometric sequences:

• a_n = a₁ * r ^{(n - 1)}*
• a₆* = 2 x 2⁵ = 2 x 32 = 64

Question 46

Find the common ratio of a geometric sequence with

a₁ = 24 and a₄ = 3.

Answer 46

The common ratio r = ¹/_2.

Use the formula for the nth term of a geometric sequence:

• a_n = a₁* r ^{(n -1)}*
• a₁* = 24 ; a₄ = 3

3 = 24 * r^(4-1) = 24 * r³

r³ = ¹/₈ ⇒ r = ¹/₂

Question 47

How many different even whole numbers are factors of

2 x 3 x 5 x 7?

Answer 47

There are 8 even whole factors:

• 2
• 2 x 3
• 2 x 5
• 2 x 7
• 2 x 3 x 5
• 2 x 5 x 7
• 2 x 3 x 7
• 2 x 3 x 5 x 7

Question 48

Which of the following numbers has the most factors of 5?

(a) 225
(b) 555
(c) 625
(d) 750

Answer 48

(c) 625

625 has 4 factors of 5. Other choices have less.

Question 49

The largest power of 4, which is a factor of 2⁴ x 5² x 4⁴, is

(a) 4
(b) 4⁴
(c) 4⁵
(d) 4⁶

Answer 49

(d) 4⁶

2⁴ = 4²

4² x 5² x 4⁴ = 5² x 4⁶

Question 50

(20) ÷ (by the product of all factors of 20) =

(a) 20
(b) ¹/₂₀
(c) ¹/₄₀₀
(d) 1

Answer 50

(c) ¹/₄₀₀

The factors of 20 are 1, 2, 4, 5, 10 and 20.

Cancel the common factors between the numerator and the denominator.

Question 51

Which of the following has the largest number of distinct prime factors?

(a) 40
(b) 42
(c) 44
(d) 46
(e) 48

Answer 51

(b) 42

42 has 3 distinct prime factors: 2, 3 and 7. The key here is to notice the word “distinct” which means different. Other choices have two different prime number factors.

Question 52

What is the least common multiple of 11, 22, 33 and 44?

(a) 66
(b) 88
(c) 99
(d) 111
(e) 132

Answer 52

(e) 132

Have you noticed that all four numbers have 11 as a common factor?

22 = 2 x 11

33 = 3 x 11

44 = 4 x 11 = 2² x 11

The LCM would be the number that is a product of 2²x 3 x 11.

Question 53

What is the largest multiple of 2 that is a factor of 64?

Answer 53

Since 64 is a multiple of 2, the largest such factor is 64.