Real Numbers and Elements of Number Theory Flashcards

This deck provides basic elements of Number Theory. It provides definitions of different type of numbers, studies the relationships between odd and even numbers, and explains prime numbers and prime factorization. At the end of the deck, there are practice questions that reinforce what you've learned as well as test your knowledge from different angles.

1
Q

What is the base of our numerical system?

A

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.

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2
Q

Name:

subsets of real numbers

A

The following are subsets of real numbers:

  • Natural numbers
  • Whole numbers
  • Integers
  • Rational numbers
  • Irrational numbers
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3
Q

Define:

natural numbers

A

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.

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4
Q

Define:

whole numbers

A

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.

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5
Q

Define:

integers

A

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.

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6
Q

Define:

rational numbers

A

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 = 5/1 1.75 = 7/4

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

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7
Q

Define:

irrational numbers

A

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.

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8
Q

Define:

even numbers

A

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.

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9
Q

Define:

odd numbers

A

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.

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10
Q

Is the sum of two even numbers even or odd?

EVEN + EVEN = ?

A

EVEN + EVEN = EVEN

Example: 10 + 2 = 12

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11
Q

Is the difference between two even numbers even or odd?

EVEN - EVEN = ?

A

EVEN - EVEN = EVEN

Example: 10 - 2 = 8

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12
Q

Is the sum of two odd numbers odd or even?

ODD + ODD = ?

A

ODD + ODD = EVEN

Example: 5 + 5 = 10

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13
Q

Is the difference between two odd numbers odd or even?

ODD - ODD = ?

A

ODD - ODD = EVEN

Example: 5 - 3 = 2

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14
Q

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

EVEN + ODD = ?

A

EVEN + ODD = ODD

ODD + EVEN = ODD

Examples:

4 + 3 = 7
5 + 4 = 9

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15
Q

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

EVEN - ODD = ?

ODD - EVEN = ?

A

EVEN - ODD = ODD

ODD - EVEN = ODD

Examples:

6 - 5 = 1
7 - 2 = 5

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16
Q

Is the product of two even numbers odd or even?

EVEN x EVEN = ?

A

EVEN x EVEN = EVEN

Example: 6 x 8 = 48

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17
Q

Is the product of two odd numbers odd or even?

ODD x ODD = ?

A

ODD x ODD = ODD

Example: 3 x 7 = 21

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18
Q

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

EVEN x ODD = ?

A

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.

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19
Q

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

(EVEN)ODD = ?

A

(EVEN)ODD = EVEN

Example: 25 = 32

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20
Q

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

(EVEN)EVEN = ?

A

(EVEN)EVEN = EVEN

Example: 44 = 256

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21
Q

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

(ODD)ODD = ?

A

(ODD)ODD = ODD

Example: 33 = 27

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22
Q

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

(ODD)EVEN = ?

A

(ODD)EVEN = ODD

Example: 72 = 49

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23
Q

True or False?

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

A

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.

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24
Q

True or False?

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

A

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.

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25
Q

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

A

ODD = EVEN + 1

Example: ODD + EVEN = EVEN + EVEN + 1

26
Q

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

A

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.

27
Q

What are consecutive numbers?

A

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…}

28
Q

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

A

To count consecutive integers, subtract the smallest from the largest and add 1.

Example:

Count the integers from 14 to 51.

51 - 14 + 1 = 38

29
Q

What type of number do you get as a result of adding different consecutive positive odd numbers?

1 + 3 = ?

1 + 3 + 5 + 7 + 9 = ?

A

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

30
Q

Define:

prime numbers

A

A prime number is a natural number greater than 1 whose only factors are itself and 1.

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}

31
Q

Define:

relatively prime numbers

A

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

32
Q

Define:

composite number

A

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

33
Q

What is prime factorization?

A

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

34
Q

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

A
  • 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

35
Q

How do you perform prime factorization by dividing by primes?

A
  • 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

36
Q

Define:

perfect square

A

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…}

37
Q

Define:

perfect cube

A

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…}

38
Q

Is zero:

  • a whole number?
  • an integer?
  • a natural number?
A

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

39
Q

Is zero

  • a positive even number?
  • an odd number?
  • or neither?
A

It is neither.

Zero is neither positive nor negative. Zero is an even number.

40
Q

Is π (Pi):

  • rational or irrational?
  • an integer?
  • even or odd?
A
  • 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
41
Q

Is the product of 355 and 828 odd or even?

*** Do not use a calculator.

A

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

42
Q

When you add 355 and 828, is the result odd or even?

*** Do not use a calculator.

A

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

43
Q

When you raise 25 to the 10th power, is the result odd or even?

2510

*** Do not use a calculator.

A

ODD

If you haven’t memorized the rule below, simply use one-digit numbers to answer the question.

(ODD)EVEN = ODD

52 = 25

44
Q

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

A

Yes, it is a rational number.

54 ÷ 8 = 6.75

45
Q

True or false:

All prime numbers are odd.

A

False.

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

46
Q

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

A

Yes, the sum of the first and second prime numbers is also a prime number.

2 + 3 = 5

47
Q

Which of the following is/are prime numbers?

29, 33, 41, 93

A

29 and 41 are prime numbers.

33 can be factored into 3 x 11. 93 can be factored into 3 x 31. Therefore, these two numbers are not prime.

48
Q

If x is a prime number, which of the following is never a prime?

(a) x + 1
(b) x + 3
(c) x + 5
(d) x + 7
(e) x + 9

A

(d) x + 7

The key here is to remember that 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.

49
Q

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

A

2 is a prime number, while neither number is composite.

*** 4 is the first composite number; it has more than 2 positive whole factors.

*** 1 is neither prime nor composite.

50
Q

What is the proper way to write the prime factorization of 126?

(a) 2 x 7 x 9
(b) 1 x 2 x 3 x 21
(c) Neither answer is correct
(d) Both answers are correct

A

(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

51
Q

What are the first and the second composite numbers?

A

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

52
Q

If a number is even, which of the following could be odd?

(a) square of the number
(b) square root of the number
(c) twice the number
(d) half the number

A

(d) half the number

Choices a), b) and c) are always even. Half of an even number could be odd.

Example: Half of 30 is 15.

53
Q

If the sum of 5 prime numbers is odd, their product could not equal

(a) 933
(b) 1,067
(c) 1,234
(d) 1,555

A

(c) 1,234

If the sum of 5 prime numbers is odd, all five are odd. The product of 5 odd numbers cannot be even.

54
Q

The sum of four odd numbers cannot equal

(a) 624
(b) 678
(c) 732
(d) 829

A

(d) 829

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

55
Q

The sum of two consecutive whole numbers cannot be

(a) 1
(b) prime
(c) composite
(d) even
(e) odd

A

(d) even

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.

56
Q

Which of the following numbers has the greatest ones digit?

(a) 333
(b) 334
(c) 335
(d) 336

Please, do not raise 3 into 36th power even if you have a calculator handy!

A

(b) 334

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.

31 = 3
32 = 9
33 = 27
34 = 81
35 = 243

57
Q

How many different positive consecutive odd numbers do you have to add together to get the value of the sum to be 625?

*** Start counting at 1.

A

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

58
Q

If the sum of 10 consecutive whole numbers is divided by 10, the remainder is always:

(a) 0
(b) 1
(c) 5
(d) 9

A

(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.

59
Q

If the sum of 20 consecutive whole numbers is divided by 10, the remainder is always

(a) 0
(b) 1
(c) 5
(d) 9

A

(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.

60
Q

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

A

The difference is even.

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

61
Q

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

A

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