Real Numbers and Elements of Number Theory

Question 1

What is the base of our numerical system?

Answer 1

Our numerical system is a decimal or base ten system. It uses digits from 0 to 9 as a base.

Our numerical system is a place-value system. This means that the place or location of a numeral determines its numerical value.

Question 2

Name:

subsets of real numbers

Answer 2

The following are subsets of real numbers:

• Natural numbers
• Whole numbers
• Integers
• Rational numbers
• Irrational numbers

Question 3

Define:

natural numbers

Answer 3

Natural numbers are the set of counting numbers.

{1, 2, 3, 4, 5…}

Natural numbers are comprised of odd and even numbers.

The smallest natural number is 1; the largest natural number is infinity.

Question 4

Define:

whole numbers

Answer 4

Whole numbers are the set of natural (counting) numbers and zero.

{0, 1, 2, 3, 4, 5…}

Whole numbers are comprised of odd and even numbers.

Question 5

Define:

integers

Answer 5

Integers are the set of natural numbers, their negative opposites, and zero.

{…-3, -2, -1, 0, 1, 2, 3…}

Integers are comprised of whole numbers and the opposites of natural numbers.

Question 6

Define:

rational numbers

Answer 6

Rational numbers are the numbers that can be expressed as simple fractions of two integers – i.e. as ratios.

*** The denominator in the fraction cannot be zero.

Examples:

5 = ⁵/₁1.75 = ⁷/₄

Rational numbers consist of integers and non-integral numbers (numbers that have terminating or repeating decimals).

Question 7

Define:

irrational numbers

Answer 7

Irrational numbers are the numbers that cannot be written as terminating or repeating decimals.

Example:

For the purposes of the SAT, the most important irrational numbers are the square root of 2, the square root of 3, and Pi.

Question 8

Define:

even numbers

Answer 8

A number that is divisible by 2 is called an even number.

{…-4, -2, 0, 2, 4…}

All numbers ending in 0, 2, 4, 6, and 8 are even.

Question 9

Define:

odd numbers

Answer 9

A number that is not divisible by 2 is called an odd number.

{…-5, -3, -1, 1, 3, 5…}

All numbers ending in 1, 3, 5, 7, and 9 are odd.

Question 10

Is the sum of two even numbers even or odd?

EVEN + EVEN = ?

Answer 10

EVEN + EVEN = EVEN

Example:

10 + 2 = 12

Question 11

Is the difference between two even numbers even or odd?

EVEN - EVEN = ?

Answer 11

EVEN - EVEN = EVEN

Example:

10 - 2 = 8

Question 12

Is the sum of two odd numbers odd or even?

ODD + ODD = ?

Answer 12

ODD + ODD = EVEN

Example:

5 + 5 = 10

Question 13

Is the difference between two odd numbers odd or even?

ODD - ODD = ?

Answer 13

ODD - ODD = EVEN

Example:

5 - 3 = 2

Question 14

Is the sum of an odd number and an even number odd or even?

EVEN + ODD = ?

Answer 14

EVEN + ODD = ODD

ODD + EVEN = ODD

Examples:

4 + 3 = 7

5 + 4 = 9

Question 15

Is the difference between an odd number and an even number odd or even?

EVEN - ODD = ?

ODD - EVEN = ?

Answer 15

EVEN - ODD = ODD

ODD - EVEN = ODD

Examples:

6 - 5 = 1

7 - 2 = 5

Question 16

Is the product of two even numbers odd or even?

EVEN x EVEN = ?

Answer 16

EVEN x EVEN = EVEN

Example:

6 x 8 = 48

Question 17

Is the product of two odd numbers odd or even?

ODD x ODD = ?

Answer 17

ODD x ODD = ODD

Example:

3 x 7 = 21

Question 18

Is the product of an odd number and an even number even or odd?

EVEN x ODD = ?

Answer 18

EVEN x ODD = EVEN

Example:

6 x 3 = 18

*** When dividing odd or even numbers, the result can be a fraction, which is not a whole number; therefore, it is neither even nor odd.

Question 19

When you raise even numbers to odd powers, is the result odd or even?

(EVEN)^ODD = ?

Answer 19

(EVEN)^ODD = EVEN

Example:

2⁵ = 32

Question 20

When you raise even numbers to even powers, is the result odd or even?

(EVEN)^EVEN = ?

Answer 20

(EVEN)^EVEN = EVEN

Example:

4⁴ = 256

Question 21

When you raise odd numbers to odd powers, is the result odd or even?

(ODD)^ODD = ?

Answer 21

(ODD)^ODD = ODD

Example:

3³ = 27

Question 22

When you raise odd numbers to even powers, is the result odd or even?

(ODD)^EVEN = ?

Answer 22

(ODD)^EVEN = ODD

Example:

7² = 49

Question 23

True or False?

Any operation (addition, subtraction, multiplication, division or raising to power) on even numbers will result in an even number.

Answer 23

True.

If you understand that any two even numbers are divisible by 2, then logically the sum, the difference, the product, the quotient, the power of the two will always be divisible by two.

Question 24

True or False?

Any operation (addition, subtraction, multiplication, division or raising to power) on odd numbers will result in an odd number.

Answer 24

False.

• The sum and the difference of two odd #’s are even
• The product, the quotient, and the power is odd

Think of ODD numbers as EVEN + 1. Or remind yourself that odd numbers end in 1, 3, 5, 7, or 9.

Example:

ODD + ODD = EVEN + EVEN + 2 = EVEN.

Question 25

How do you express an odd number in terms of an even number?

Answer 25

ODD = EVEN + 1 *Example:* ODD + EVEN = EVEN + EVEN + 1

Question 26

How should you use number facts like ODD + ODD = EVEN on the SAT test?

Answer 26

You don't have to memorize them, but you have to be able to see that some questions may need you to recall and connect these facts to solve quickly. Remember, SAT type questions often use simple facts, and the trick is seeing through them for a quick solution.

Question 27

What are **consecutive** numbers?

Answer 27

**Consecutive** numbers follow the natural order and differ by 1. {...4, 5, 6, 7, 8, 9...} Consecutive even and consecutive odd numbers differ by 2. {...2, 4, 6, 8...} {...3, 5, 7, 9...}

Question 28

In a set of *consecutive* integers, how do you find the number of integers between the smallest and the largest numbers, inclusively?

Answer 28

To count consecutive integers, *subtract* the smallest from the largest and *add 1*. *Example*: Count the integers from 14 to 51. 54 - 14 + 1 = 41

Question 29

What type of number do you get as a result of adding *different consecutive positive odd* numbers? 1 + 3 = ? 1 + 3 + 5 + 7 + 9 = ?

Answer 29

The sum is a *perfect square* of the number of numbers being added together. 1 + 3 = 4 = 2² 1 + 3 + 5 + 7 + 9 = 25 = 5²

Question 30

# Define: **prime** numbers

Answer 30

A **prime number** is a natural number *greater than 1* whose only factors are *itself* and *1*. ## Footnote The following is a set of prime numbers less than 100: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}

Question 31

# Define: **relatively prime** numbers

Answer 31

Two **relatively prime** numbers are natural numbers that have no common factors other than 1. To determine if two numbers are relatively prime, do prime factorization on them. *Example:* 25 and 28 are relatively prime because they have no prime factors in common. 25 = 5 x 5 28 = 7 x 2 x 2

Question 32

# Define: **composite** number

Answer 32

A **composite number** is any positive integer *greater than 1* that is *not* a prime number. {4, 6, 8, 10, 12, 14, 15...} * The *first* composite number is 4 * Every integer greater than 1 is either a prime number or a composite number * 1 is neither prime nor composite

Question 33

What is **prime factorization**?

Answer 33

**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 34

How do you use a **factor tree** to perform prime factorization?

Answer 34

* Find any pair of factors of the number * Circle the prime factor(s) * Find any other factors for the non-prime factor * Repeat the process until you find all the prime factors * Put them together as a product using exponents *Example:* 2² x 3 x 7 x 13

Question 35

How do you perform prime factorization by **dividing by primes**?

Answer 35

* Divide by 2 if possible, until the quotient is no longer divisible by 2 * Divide by 3 the same way * Continue this pattern, dividing by prime numbers only, until the quotient is prime * Write the product of the divisors and the quotient using exponents *Example*: Express 72 as a product of prime numbers using the dividing by primes method. 72 ÷ 2 = 36 36 ÷ 2 = 18 18 ÷ 2 = 9 9 ÷ 3 = 3 72 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²

Question 36

# Define: **perfect square**

Answer 36

A **perfect square** is any integer that is a square of another integer. *Example:* 3² = 9 Below is a set of perfect squares from 1 to 100: {1, 4, 9, 16, 25, 36, 49, 64, 81, 100...}

Question 37

# Define: **perfect cube**

Answer 37

A **perfect cube** is any integer that is a cube of another integer. *Example:* 4³ = 4 x 4 x 4 = 64 Below is a set of perfect cubes from 1 to 1000: {1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000...}

Question 38

Is **zero**: * a whole number? * an integer? * a natural number?

Answer 38

Zero is a whole number and an integer but it is *not* a natural number.

Question 39

Is **zero** * a positive even number? * an odd number? * or neither?

Answer 39

It is neither. ## Footnote Zero is neither positive nor negative. Zero is an even number.

Question 40

Is *π* (Pi): * rational or irrational? * an integer? * even or odd?

Answer 40

* *π* (Pi) is an *irrational* number. Its decimals never terminate or repeat * It *cannot* be an integer because it is an irrational number * It's neither even nor odd. Even or odd is a property of whole numbers

Question 41

Is the product of 355 and 828 odd or even? ## Footnote \*\*\* Do not use a calculator.

Answer 41

EVEN Don't multiply 355 by 828. If you haven't memorized the rule below, simply use any one-digit odd and even numbers. ODD x EVEN = EVEN 5 x 8 = 40

Question 42

When you add 355 and 828, is the result odd or even? ## Footnote \*\*\* Do not use a calculator.

Answer 42

ODD Don't add 355 and 828 to answer the question. If you haven't memorized the rule below, simply add unit digits of the two numbers together. If their sum is an odd number, the sum of 355 and 828 is an odd number as well. ODD + EVEN = ODD 5 + 8 =13

Question 43

When you raise 25 to the 10^th power, is the result odd or even? 25^{10 *** Do not use a calculator.}

Answer 43

ODD If you haven't memorized the rule below, simply use one-digit numbers to answer the question. (ODD)^EVEN = ODD 5² = 25

Question 44

When 54 is divided by 8, is the result a rational number?

Answer 44

Yes, it is a rational number. 54 ÷ 8 = 6.75

Question 45

True or false: all prime numbers are odd.

Answer 45

False. The first prime number is 2, which is the only even prime number.

Question 46

Is the sum of the first and second prime numbers a prime number itself?

Answer 46

Yes, the sum of the first and second prime numbers is also a prime number. 2 + 3 = 5

Question 47

Which of the following is/are prime numbers? 29, 33, 41, 93

Answer 47

29 and 41 are prime numbers. ## Footnote 33 can be factored into 3 x 11. 93 can be factored into 3 x 31. Therefore, these two numbers are not prime.

Question 48

If *x* is a prime number, which of the following is *never* a prime? ## Footnote (a) *x* + 1 (b) *x* + 3 (c) *x* + 5 (d) *x* + 7 (e) *x* + 9

Answer 48

(d) *x* + 7 ## Footnote The key here is to remember than the first prime number is 2 and it's different from all other prime numbers. It's even. So, pick 2 and 3 to plug into the answer choices.

Question 49

Of the numbers 1 and 2, which is a prime number and which is a composite number?

Answer 49

2 is a prime number, while *neither* number is composite. ## Footnote \*\*\* 4 is the first composite number; it has more than 2 positive whole factors. \*\*\* 1 is neither prime nor composite.

Question 50

What is the proper way to write the prime factorization of 126? ## Footnote (a) 2 x 7 x 9 (b) 1 x 2 x 3 x 21 (c) Neither answer is correct (d) Both answers are correct

Answer 50

(c) Neither answer is correct. 1, 9, and 21 are *not* prime numbers. 2 x 3 x 3 x 7 is the correct way to factor 126 into prime factors. Or, using exponents: 2 x 3² x 7

Question 51

What are the first and the second composite numbers?

Answer 51

The first composite number is 4 and the second is 6.

Question 52

If a number is *even*, which of the following *could* be odd? ## Footnote (a) square of the number (b) square root of the number (c) twice the number (d) half the number

Answer 52

(d) half the number ## Footnote Choices a), b) and c) are always even. Half of an even number could be odd. *Example*: Half of 30 is 15.

Question 53

If the sum of 5 prime numbers is *odd*, their product could *not* equal ## Footnote (a) 933 (b) 1,067 (c) 1,234 (d) 1,555

Answer 53

(c) 1,234 ## Footnote If the sum of 5 prime numbers is odd, *all five are odd*. The product of 5 odd numbers *cannot* be even.

Question 54

The sum of four odd numbers *cannot* equal ## Footnote (a) 624 (b) 678 (c) 732 (d) 829

Answer 54

(d) 829 ## Footnote The sum of an even number of odd numerals *cannot* be an odd number. *ODD + ODD + ODD + ODD* = ODD x 4 = (EVEN + 1) x 4 ⇒ 4 EVEN + 4 = *EVEN*

Question 55

The sum of two *consecutive* whole numbers *cannot* be ## Footnote (a) 1 (b) prime (c) composite (d) even (e) odd

Answer 55

(d) even ## Footnote A pair of any consecutive numbers consists of one odd and one even number. The sum of an odd and an even number is always odd.

Question 56

Which of the following numbers has the greatest *ones* digit? ## Footnote (a) 3³³ (b) 3³⁴ (c) 3³⁵ (d) 3³⁶ Please, do not raise 3 into 36th power even if you have a calculator handy!

Answer 56

(b) 3³⁴ ## Footnote The ones digit cycle is 3, 9, 7, 1, 3..... It repeats every 4. Every 2nd number in this cycle ends in 9. 3³⁴ is the second number in the cycle and therefore, it has the greatest ones digit. ``` 3<sup>1</sup> = 3 3<sup>2</sup> = 9 3<sup>3</sup> = 27 3<sup>4</sup> = 81 3<sup>5</sup> = 243 ```

Question 57

How many *different positive consecutive odd* numbers do you have to add together to get the value of the sum to be 625? ## Footnote \*\*\* Start counting at 1.

Answer 57

You need to add 25 consecutive odd numbers for their sum to be 625. When you add consecutive odd numbers, the sum is a *perfect square* of the amount of numbers being added together. *Example:* 1 + 3 + 5 + 7 = 16 = 4²

Question 58

If the sum of 10 consecutive whole numbers is divided by 10, the remainder is always: ## Footnote (a) 0 (b) 1 (c) 5 (d) 9

Answer 58

(c) 5 You can manually add numbers from 1 to 10 and divide by 10. Or add these numbers in pairs: (10 + 1), (9 + 2), (8 + 3), etc. There are 5 pairs. The sum of each pair is 11. To find the sum of all numbers between 1 and 10 multiply 11 x 5 = 55. Now, divide by 10.

Question 59

If the sum of *20* consecutive whole numbers is divided by 10, the remainder is always ## Footnote (a) 0 (b) 1 (c) 5 (d) 9

Answer 59

(a) 0 You can add numbers from 1 to 20 to find the answer but it will take a long time. Better, add these numbers in pairs: (20 + 1), (19 + 2), (18 + 3), etc. The sum of each of 10 pairs is 21. To find the sum of all numbers between 1 and 20, multiply 21 x 10. The result is 210 and it is divisible by 10 without a remainder.

Question 60

Is the difference between any two multiples of an *odd* number even or odd?

Answer 60

The difference is even. ## Footnote If the *product* of the two multiples is *odd*, the *multiples* must be *odd*. The difference of the two odd numbers is always even.

Question 61

How many two-digit and three-digit counting numbers are there?

Answer 61

There are 990 two-digit and three-digit counting numbers. The first two-digit number is 10. The last three-digit number is 999. 999 - 10 + 1 = 990