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Flashcards in Real Numbers and Elements of Number Theory Deck (61)
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31
Define:

relatively prime numbers

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
Define:

composite number

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

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 = 25 x 3

34

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

  • 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

How do you perform prime factorization by dividing by primes?

  • 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
Define:

perfect square

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
Define:

perfect cube

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

Is zero:

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

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

39

Is zero

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

It is neither.

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

40

Is π (Pi):

  • rational or irrational?
  • an integer?
  • even or odd?

  • 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

Is the product of 355 and  828 odd or even?

*** Do not use a calculator.

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

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

*** Do not use a calculator.

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

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

2510

*** Do not use a calculator.

ODD

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

(ODD)EVEN = ODD

52 = 25

44

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

Yes, it is a rational number.

54 ÷ 8 = 6.75

45

True or false:

all prime numbers are odd.

False.

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

46

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

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

2 + 3 = 5

47

Which of the following is/are prime numbers?

29, 33, 41, 93

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

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

(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

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

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

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

(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

What are the first and the second composite numbers?

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

52

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

(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

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

(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

The sum of four odd numbers cannot equal

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

(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

The sum of two consecutive whole numbers cannot be

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

(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

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!

(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

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.

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

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

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

(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

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) 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

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

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.