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Fractions!!. Prime Numbers and Factorization. How many positive prime numbers are less than 100?. Answer: 25. What is the sum of the prime numbers between π and 10 π ?. Answer: 155. List the positive prime numbers less than 100 that have the units digit equal to 3.

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Prime numbers and factorization

Prime NumbersandFactorization







Answer 3 13 23 43 53 73 83
Answer: units digit equal to 3.3, 13, 23, 43, 53, 73, 83


Prime factorization

Prime Factorization units digit equal to 3.


Factoring units digit equal to 3. is like taking a number apart. It means to express a number as the product of its factors. Factors are either composite numbers or prime numbers (except that 0 and 1 are neither prime nor composite).The number 12 is a multiple of 3, because it can be divided evenly by 3.3 · 4 = 123 and 4 are both factors of 123 · 2 · 2 (prime factorization of 12)12 is a multiple of both 3 and 4.


Write the prime factorization of: units digit equal to 3.

100


Answer 2 2 5 5 or 2 5 prime power factorization
Answer: units digit equal to 3.2∙2∙5∙5or2² ∙ 5²(Prime-Power Factorization)


Write the prime factorization of: units digit equal to 3.

972


Answer 2 2 3 3 3 3 3 or 2 3 prime power factorization
Answer: units digit equal to 3.2∙2∙3∙3∙3∙3∙3 or2² ∙ 3⁵(Prime-Power Factorization)



Answer 5 5 n n
Answer: (includes variables):5·5·n·n



Answer 2 2 23 x y
Answer: (includes variables):2·2·23·x·y


Factors

Factors (includes variables):


List all the positive factors of (includes variables):

30


Answer 1 2 3 5 6 10 15 30
Answer: (includes variables):1, 2, 3, 5, 6, 10, 15, 30


List all the positive factors of (includes variables):

87


Answer 1 3 29 87
Answer: (includes variables):1, 3, 29, 87


Greatest common factor gcf

Greatest Common Factor (GCF) (includes variables):


Greatest Common Factor (includes variables):The highest number that divides exactly into two or more numbers. If you find all the factors of two or more numbers, and some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.Abbreviated "GCF". Also called "Highest Common Factor"Example: the GCF of 12 and 30 is 6, because 1, 2, 3 and 6 are factors of both 12 and 30, and 6 is the greatest.


Find the GCF of (includes variables):

24 & 28


Answer 4
Answer: (includes variables):4


Find the GCF of (includes variables):

39v & 30uv


Answer 3v
Answer: (includes variables):3v


Find the GCF of (includes variables):

35n²m & 21m²n


Answer 7nm
Answer: (includes variables):7nm


Find the GCF of (includes variables):

36xy³ & 24y²


Answer 12y
Answer: (includes variables):12y²


Find the GCF of (includes variables):

105x, 30yx & 75x


Answer 15x
Answer: (includes variables):15x


Least common multiple lcm

Least Common Multiple (LCM) (includes variables):


Least Common Multiple (includes variables):The smallest (non-zero) number that is a multiple of two or more numbers.Least Common Multiple is made up of the words Least, Common and Multiple:What is a "Multiple" ?The multiples of a number are what you get when you multiply it by other numbers (such as if you multiply it by 1,2,3,4,5, etc). Just like the multiplication table. Here are some examples: The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, etc ... The multiples of 12 are: 12, 24, 36, 48, 60, 72, etc... What is a "Common Multiple" ?When you list the multiples of two (or more) numbers, and find the same value in both lists, then that is a common multiple of those numbers. For example, when you write down the multiples of 4 and 5, the common multiples are those that are found in both lists:The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,... The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,...Notice that 20 and 40 appear in both lists? So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc ..., too) What is the "Least Common Multiple" ?It is simply the smallest of the common multiples. In our previous example, the smallest of the common multiples is 20 ...... so the Least Common Multiple of 4 and 5 is 20.


Finding the Least Common Multiple (includes variables):It is a really easy thing to do. Just start listing the multiples of the numbers until you get a match.Example: Find the least common multiple for 3 and 5:The multiples of 3 are 3, 6, 9, 12, 15, ..., and the multiples of 5 are 5, 10, 15, 20, ..., like this:As you can see on this number line, the first time the multiples match up is 15. Answer: 15More than 2 NumbersYou can also find the least common multiple of 3 (or more) numbers.Example: Find the least common multiple for 4, 6, and 8Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...Multiples of 6 are: 6, 12, 18, 24, 30, 36, ...Multiples of 8 are: 8, 16, 24, 32, 40, ....So, 24 is the least common multiple (I can't find a smaller one !)


Find the LCM of (includes variables):

35 & 25


Answer 175
Answer: (includes variables):175


Find the LCM of (includes variables):

30, 25 & 10


Answer 150
Answer: (includes variables):150


Find the LCM of (includes variables):

18 & 6v


Answer 18v
Answer: (includes variables):18v


Find the LCM of (includes variables):

3x² & 10


Answer 30x
Answer: (includes variables):30x²


Find the LCM of (includes variables):

20y & 14y²


Answer 140y
Answer: (includes variables):140y²


Find the LCM of (includes variables):

16x²y & 32x


Answer 32x y
Answer: (includes variables):32x²y


Find the LCM of (includes variables):

8y², 16xy & 16y


Answer 16y x
Answer: (includes variables):16y²x


What is the negative difference of the least common multiple (LCM) and the greatest common factor (GCF) of:

80, 140, and 200


Answer 2780
Answer: (LCM) and the greatest common factor (GCF) of:-2780


Types of fractional numbers
Types of Fractional Numbers (LCM) and the greatest common factor (GCF) of:

  • A proper fraction is a fraction whose value is less than 1 (numerator < denominator)

  • An improper fraction is a fraction whose value is greater than or equal to 1 (numerator > denominator)

  • A mixed number is a number whose value is greater than 1 made up of a whole part and a fraction part


Improper fraction mixed number
Improper Fraction (LCM) and the greatest common factor (GCF) of:Mixed Number

  • Denominator: tells us how many parts make up a whole

  • Numerator: tells us how many parts we have

  • How many wholes can we make out of the parts we have?

  • Divide the numerator by the denominator  the quotient is the whole part

  • How many parts do we have remaining?

  • The remainder (over the denominator) makes up the fraction part


Simplest form of a fraction
Simplest Form of a Fraction (LCM) and the greatest common factor (GCF) of:

  • A fraction is in simplest form when there are no common factors in the numerator and the denominator.


Ex simplest form
Ex: Simplest Form (LCM) and the greatest common factor (GCF) of:

Ex: 6/8 and 3/4 are equivalent

The fraction 6/8 is written in simplest form as 3/4

=

=

=

1 x

Magic one


Ex write 12 42 in simplest form
Ex: Write 12/42 in simplest form (LCM) and the greatest common factor (GCF) of:

  • First prime factor the numerator and the denominator:

  • 12 = 2 x 2 x 3 and 42 = 2 x 3 x 7

  • Look for Magic Ones

  • Simplify

=

=

=

1 x 1 x

=

Notice: 2 x 3 = 6 = GCF(12, 42)

 factoring (dividing) out the GCF will simplify the fraction


Ex write 7 28 in simplest form
Ex: Write 7/28 in simplest form (LCM) and the greatest common factor (GCF) of:

  • What is the GCF(7, 28)?

    • Hint: prime factor 7 = 7

    • prime factor 28 = 2 x 2 x 7

= 7

=

=

=

1 x

=

Dividing out the GCF from the numerator and denominator simplifies the fraction.


Ex write 27 56 in simplest form
Ex: Write 27/56 in simplest form (LCM) and the greatest common factor (GCF) of:

  • What is the GCF(27, 56)?

    • Hint: prime factor 27 = 3 x 3 x 3

    • prime factor 56 = 2 x 2 x 2 x 7

= 1

There is no common factor to the numerator and denominator (other than 1)

Therefore, 27/56 is in simplest form.


Rational and Irrational Numbers (LCM) and the greatest common factor (GCF) of:


Rational and Irrational Numbers Essential Question (LCM) and the greatest common factor (GCF) of:

How do I distinguish between rational and irrational numbers?


Real Numbers (LCM) and the greatest common factor (GCF) of:

Rational numbers

Irrational numbers

Integers

Whole

numbers

The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.


Rational (LCM) and the greatest common factor (GCF) of:numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.

⅔, ⅕, ¼ or

144 = 12


Caution! (LCM) and the greatest common factor (GCF) of:

A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

Irrational numberscan be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, the square root of 2 is not a perfect square, so the square root of 2 is irrational. Also, π is irrational.


Make a Venn Diagram that displays the following sets of numbers:

Reals, Rationals, Irrationals, Integers, Wholes, and Naturals.

Reals

Rationals

-2.65

Integers

-3

-19

Wholes

0

Irrationals

Naturals

1, 2, 3...


C numbers:lassifying Real Numbers

Write all classifications that apply to each number.

A.

5 is a whole number that is not a perfect square.

5

irrational, real

B.

–12.75

–12.75 is a terminating decimal.

rational, real

16

C.

whole, integer, rational, real


9 = 3 numbers:

Write all classifications that apply to each number.

9

A.

whole, integer, rational, real

–35.9

–35.9 is a terminating decimal.

B.

rational, real

81

C.

whole, integer, rational, real


A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.


Adding and subtracting fractions

Adding and Subtracting cannot divide by zero. So it is not a number at all.Fractions


Evaluate the expression 6
Evaluate the expression: cannot divide by zero. So it is not a number at all.6- ¹⁄₆


Answer
Answer: cannot divide by zero. So it is not a number at all.³⁵/₆


Evaluate the expression
Evaluate the expression: cannot divide by zero. So it is not a number at all.⅕+ ¾


Answer1
Answer: cannot divide by zero. So it is not a number at all.¹⁹/₂₀


Evaluate the expression 4 5 7 8
Evaluate the expression: cannot divide by zero. So it is not a number at all.(-4/5) – 7/8


Answer2
Answer: cannot divide by zero. So it is not a number at all.- ⁶⁷/₄₀


Evaluate the expression 10 7 1 6
Evaluate the expression: cannot divide by zero. So it is not a number at all.(-10/7) + 1/6


Answer3
Answer: cannot divide by zero. So it is not a number at all.- ⁵³/₄₂


Evaluate the expression 2
Evaluate the expression: cannot divide by zero. So it is not a number at all.2 – ¹³/₈


Answer 3 8
Answer: cannot divide by zero. So it is not a number at all.3/8


Evaluate the expression 4 3 3 2
Evaluate the expression: cannot divide by zero. So it is not a number at all.(-4/3) – (-3/2)


Answer 1 6
Answer: cannot divide by zero. So it is not a number at all.1/6


Evaluate the expression 3 4
Evaluate the expression: cannot divide by zero. So it is not a number at all.- 3 ⅗ - 4 ⅖


Answer 8
Answer: cannot divide by zero. So it is not a number at all.-8


Evaluate the expression 2 1
Evaluate the expression: cannot divide by zero. So it is not a number at all.(-2⁷/₈) + (-1 ¹/₂)


Answer 41
Answer: cannot divide by zero. So it is not a number at all. -4 ³/₈


Evaluate the expression 2 11
Evaluate the expression: cannot divide by zero. So it is not a number at all.(-2⁵/₆) - (-1 ¹/₄)


Answer 1
Answer: cannot divide by zero. So it is not a number at all. -1 ⁷/₁₂


Evaluate the expression 21
Evaluate the expression: cannot divide by zero. So it is not a number at all.2⁴/₅ - ⁵/₈


Answer 2
Answer: cannot divide by zero. So it is not a number at all. 2 ⁷/₄₀


Multiplying fractions
Multiplying Fractions cannot divide by zero. So it is not a number at all.


Evaluate the expression1
Evaluate the expression: cannot divide by zero. So it is not a number at all.-⁵/₄ · ¹/₃


Answer4
Answer: cannot divide by zero. So it is not a number at all. -⁵/₁₂


Evaluate the expression 22
Evaluate the expression: cannot divide by zero. So it is not a number at all.-2 · ³/₇


Answer5
Answer: cannot divide by zero. So it is not a number at all. -⁶/₇


Dividing fractions
Dividing Fractions cannot divide by zero. So it is not a number at all.


Evaluate the expression2
Evaluate the expression: cannot divide by zero. So it is not a number at all.- ¹/₅ ÷ ⁷/₄


Answer6
Answer: cannot divide by zero. So it is not a number at all. - ⁴/₃₅


Evaluate the expression3
Evaluate the expression: cannot divide by zero. So it is not a number at all.- ¹/₂ ÷ ⁵/₄


Answer7
Answer: cannot divide by zero. So it is not a number at all. - ²/₅


Evaluate the expression4
Evaluate the expression: cannot divide by zero. So it is not a number at all.- ³/₂ ÷ ⁻¹⁰/₇


Answer8
Answer: cannot divide by zero. So it is not a number at all. ²¹/₂₀


Evaluate the expression 23
Evaluate the expression: cannot divide by zero. So it is not a number at all.- ⁹/₅ ÷ 2


Answer9
Answer: cannot divide by zero. So it is not a number at all. -⁹/₁₀


Multiplying and dividing mixed numbers
Multiplying and Dividing cannot divide by zero. So it is not a number at all.Mixed Numbers


Evaluate the expression 2 4
Evaluate the expression: cannot divide by zero. So it is not a number at all.-2²/₃ · 4 ¹/₁₀


Answer 10
Answer: cannot divide by zero. So it is not a number at all.-10 ¹⁴/₁₅


Evaluate the expression 2 12
Evaluate the expression: cannot divide by zero. So it is not a number at all.-2 ¹/₅ · (-1 ³/₄)


Answer 3
Answer: cannot divide by zero. So it is not a number at all. 3 ¹⁷/₂₀


Evaluate the expression 2 3
Evaluate the expression: cannot divide by zero. So it is not a number at all.-2 ÷ (-3 ⁴/₅)


Answer10
Answer: cannot divide by zero. So it is not a number at all.+ ¹⁰/₁₉


Evaluate the expression 3 2
Evaluate the expression: cannot divide by zero. So it is not a number at all.- 3 ⁷/₁₀ ÷ 2 ¹/₄


Answer 11
Answer: cannot divide by zero. So it is not a number at all.-1 ²⁹/₄₅


Challenge Problems cannot divide by zero. So it is not a number at all.


3 3 2 3 2
3 cannot divide by zero. So it is not a number at all._________ 3___ 2 + 3/2

2 +

2 +


Answer 31
Answer: cannot divide by zero. So it is not a number at all.3 ¹/₂₀


Simplify completely: cannot divide by zero. So it is not a number at all.

4x²y

2x


Answer 2xy
Answer: cannot divide by zero. So it is not a number at all.2xy


Simplify completely: cannot divide by zero. So it is not a number at all.

y¯¹

y¯²


Answer y
Answer: cannot divide by zero. So it is not a number at all.y


Simplify completely: cannot divide by zero. So it is not a number at all.

16x⁴y¯¹

4x²y¯²


Answer 4x y
Answer: cannot divide by zero. So it is not a number at all.4x²y


Simplify completely: cannot divide by zero. So it is not a number at all.

36x³y⁶z¹²

4x¯¹y³z¹⁰


Answer 9x y z
Answer: cannot divide by zero. So it is not a number at all.9x⁴y³z²


Simplify completely: cannot divide by zero. So it is not a number at all.

21x³y⁷z¹⁴30x³z¯⁵

18x⁴y⁶ y¹²z¯⁶

·


Answer 35x z y
Answer: cannot divide by zero. So it is not a number at all.35x²z¹⁵y¹¹


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