Greatest Common Factor 5-5
If we list the factors of 30 and 42, we notice
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
We notice that 1, 2, 3, and 6 are all COMMON factors of these two numbers. The number 6 is the greatest of these and therefore is called the
GREATEST COMMON FACTOR of the two numbers. We write
GCF(30,42) = 6
Although listing the factors of two numbers and then comparing their common factors is one way to determine the greatest common factor, using prime factorization is another easy way to find the GCF
Find GCF(54, 72)
54 = 2 ⋅ 3 ⋅ 3 ⋅ 3
72 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3
Find the greatest power of 2 that occurs IN BOTH prime factorization. The greatest power of 2 that occurs in both is just 2 1
Find the greatest power of 3 that occurs IN BOTH prime factorizations. The greatest power of 3 that occurs in both is 32
Therefore
GCF(54, 72) = 2 ⋅ 32 = 18
In class we circled the common factors and realized that
GCF(54, 72) = 2 ⋅ 3 ⋅ 3 = 18
Fin the GCF( 45, 60)
45 = 3 ⋅ 3⋅ 5
60 = 2⋅ 2⋅ 3⋅ 5
Since 2 is NOT a factor of 45-- there is NO greatest power of 2 that occurs in both prime factorizations.
The greatest power of 3 is just 31
and the greatest power of 5 is just 51
Therefore,
GCF(45,60) = 3⋅ 5 = 15
The number 1 is a common factor of any two whole numbers!! If 1 is the GCF , then the two numbers are said to be RELATIVELY PRIME. Two numbers can be relatively prime even if one or both of them are composite.
Show that 15 and 16 are relatively prime
List the factors of each number
FACTORS of 15: 1, 3, 5, 15
FACTORS of 16: 1, 2, 4, 8, 16
Since the GCF(15,16) = 1. The two numbers are relatively prime!!
Friday, December 2, 2011
Wednesday, November 30, 2011
Algebra Honors (Period 6 & 7)
Multiplying Fractions 6-2
You know from previous years that
ac/bd = a/b ⋅ c/d
and you know the converse is also true
a/b ⋅ c/d = ac/bd
That means you could solve
8/9⋅3/10 by either multiplying first and then simplify or you could simplify first and then multiply.
I find it works so much better to simplify first
8/9⋅3/10 = 4/15
6x/y3⋅y2/15 = 2x/5y where y ≠0
Which simplifies to
This textbook wants us to keep the factored form as our answers--> so let's continue to do that. In addition it states, " ...from now on, assume that the domains of the variables do not include values for which any denominator is ZERO. Therefore it will NOT be necessary to show the excluded [or restrictions] values of the variables."
I know everyone is jumping for joy!!
Rule of Exponents for a Power of a Quotient
(a/b)m = am/bm
(x/3)3 = x3/27
(-c/2)2⋅4/3c
you must do the exponent portion first!!
c2/4⋅(4/3c)
c/3
Find the volume of a cube if each edge has length 6n/7 in
You just need to cube each factor
(6n/7)3 = 216n3/343 inches cubed
If you traveled for 7t/60 hours at 80r/9 mi/h, how far have you gone?
Just multiply
7t/60⋅80r/9
but simplify first and you get
28rt/27 miles
You know from previous years that
ac/bd = a/b ⋅ c/d
and you know the converse is also true
a/b ⋅ c/d = ac/bd
That means you could solve
8/9⋅3/10 by either multiplying first and then simplify or you could simplify first and then multiply.
I find it works so much better to simplify first
8/9⋅3/10 = 4/15
6x/y3⋅y2/15 = 2x/5y where y ≠0
Which simplifies to
This textbook wants us to keep the factored form as our answers--> so let's continue to do that. In addition it states, " ...from now on, assume that the domains of the variables do not include values for which any denominator is ZERO. Therefore it will NOT be necessary to show the excluded [or restrictions] values of the variables."
I know everyone is jumping for joy!!
Rule of Exponents for a Power of a Quotient
(a/b)m = am/bm
(x/3)3 = x3/27
(-c/2)2⋅4/3c
you must do the exponent portion first!!
c2/4⋅(4/3c)
c/3
Find the volume of a cube if each edge has length 6n/7 in
You just need to cube each factor
(6n/7)3 = 216n3/343 inches cubed
If you traveled for 7t/60 hours at 80r/9 mi/h, how far have you gone?
Just multiply
7t/60⋅80r/9
but simplify first and you get
28rt/27 miles
Math 6 Honors ( Periods 1, 2, & 3)
Prime Numbers & Composite Numbers 5-4
A prime number is one that has only two factors: 1 and the number itself, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
A counting number that has more than two factors is called a composite number, such as 4, 6, 8, 9, 10...
Since one has exactly ONE factor, it is NEITHER PRIME NOR COMPOSITE!!
Zero is also NEITHER PRIME NOR COMPOSITE!!
Sieve of Eratosthenes - We did it!! :)
Every counting number greater than 1 has at least one prime factor -- which may be the number itself.
You can factor a number into PRIME FACTORS by using a factor tree or the inverted division, as shown in class.
Using the inverted division, you also start with the smallest prime number that is a factor... and work down
give the prime factors of 42
2⎣42
3⎣21
7
When we write 42 as 2⋅3⋅7 this product of prime factors is called the prime factorization of 42.
Two is the only even prime number because all the other even numbers have two as a factor.
Explain how you know that each of the following numbers must be composite...
111; 111,111; 111,111,111; and so on....
Using your divisibility rules you notice that the sums of the digits are multiples of 3.
List all the possible digits that can be the last digit of a prime number that is greater than 10.
1, 3, 7, 9.
Choose any six digit number such that the last three digits are a repeat of the first three digits. For example
652,652. You will find that 7, 11, and 13 are all factors of that number... no matter what number you choose... why is that???? email me your response.
A prime number is one that has only two factors: 1 and the number itself, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
A counting number that has more than two factors is called a composite number, such as 4, 6, 8, 9, 10...
Since one has exactly ONE factor, it is NEITHER PRIME NOR COMPOSITE!!
Zero is also NEITHER PRIME NOR COMPOSITE!!
Sieve of Eratosthenes - We did it!! :)
Every counting number greater than 1 has at least one prime factor -- which may be the number itself.
You can factor a number into PRIME FACTORS by using a factor tree or the inverted division, as shown in class.
Using the inverted division, you also start with the smallest prime number that is a factor... and work down
give the prime factors of 42
2⎣42
3⎣21
7
When we write 42 as 2⋅3⋅7 this product of prime factors is called the prime factorization of 42.
Two is the only even prime number because all the other even numbers have two as a factor.
Explain how you know that each of the following numbers must be composite...
111; 111,111; 111,111,111; and so on....
Using your divisibility rules you notice that the sums of the digits are multiples of 3.
List all the possible digits that can be the last digit of a prime number that is greater than 10.
1, 3, 7, 9.
Choose any six digit number such that the last three digits are a repeat of the first three digits. For example
652,652. You will find that 7, 11, and 13 are all factors of that number... no matter what number you choose... why is that???? email me your response.
Math 6 Honors ( Periods 1, 2, & 3)
Prime Numbers and Composite Numbers 5-4
A prime number is a positive integer greater than 1 with exactly two factors, 1 and the number itself. The numbers 2, 3, 5, 7 are examples of prime numbers
A composite number is a positive integer greater than 1 with more than two factors. The numbers 4, 6, 8, 9, and 10 are examples of composite numbers.
Since 1 has exactly 1 factor, it is neither prime nor composite.
About 230 BCE Erathosthenes, a Greek Mathematician suggested a way to find prime numbers—up to a specific number. The method is called the Sieve of Eratosthenes because it picks out the prime numbers as a strainer, or sieve, picks out solid particles from a liquid.
You may factor a number into prime factors by using either of the following methods
➢ Inverted short division
➢ Factor tree
Both were shown in class.
Could you start the factor tree differently? If so, would you end up with the same answer?
The prime factors of 42 are the same in either factor tree, except for their order.
Every composite number greater than 1 can be written as a product of prime factors in exactly one way, except for the order of the factors.
When we write 42 as 2 ∙ 3 ∙ 7 this product is called the prime factorization of 42
Notice the order in which prime factorization is written.
Let’s try finding the prime factorization of 60
The prime factorization of 60 = 2 ∙ 2∙ 3 ∙ 5 or 22∙ 3∙ 5
A prime number is a positive integer greater than 1 with exactly two factors, 1 and the number itself. The numbers 2, 3, 5, 7 are examples of prime numbers
A composite number is a positive integer greater than 1 with more than two factors. The numbers 4, 6, 8, 9, and 10 are examples of composite numbers.
Since 1 has exactly 1 factor, it is neither prime nor composite.
About 230 BCE Erathosthenes, a Greek Mathematician suggested a way to find prime numbers—up to a specific number. The method is called the Sieve of Eratosthenes because it picks out the prime numbers as a strainer, or sieve, picks out solid particles from a liquid.
You may factor a number into prime factors by using either of the following methods
➢ Inverted short division
➢ Factor tree
Both were shown in class.
Could you start the factor tree differently? If so, would you end up with the same answer?
The prime factors of 42 are the same in either factor tree, except for their order.
Every composite number greater than 1 can be written as a product of prime factors in exactly one way, except for the order of the factors.
When we write 42 as 2 ∙ 3 ∙ 7 this product is called the prime factorization of 42
Notice the order in which prime factorization is written.
Let’s try finding the prime factorization of 60
The prime factorization of 60 = 2 ∙ 2∙ 3 ∙ 5 or 22∙ 3∙ 5
Tuesday, November 29, 2011
Math 6 Honors ( Periods 1, 2, & 3)
Square Numbers and Square Roots 5-3
Numbers such as 1, 4, 9, 16, 25, 36, 49... are called square numbers or PERFECT SQUARES.
One of two EQUAL factors of a square is called the square root of the number. To denote a square root of a number we use a radical sign (looks like a check mark with an extension) See our textbook page 157.
Although we use a radical sign to denote cube roots, fourth roots and more, without a small number on the radical sign, we have come to call that the square root.
SQRT = stands for square root, since this blog will not let me use the proper symbol) √ is the closest to the symbol
so the SQRT of 25 is 5. Actually 5 is the principal square root. Since 5 X 5 = 25
There is another root because
(-5)(-5) = 25 but in this class we are primarily interested in the principal square root or the positive square root.
Evaluate the following:
SQRT 36 + SQRT 64 = 6 + 8 = 14
SQRT 100 = 10
Is it true that SQRT 36 + SQRT 64 = SQRT 100? No
You cannot add square roots in that manner.
However look at the following:
Evaluate
SQRT 225 = 15
(SQRT 9)(SQRT 25)= (3)(5) = 15
so
SQRT 225 = (SQRT 9)(SQRT 25)
Also notice that the SQRT 1600 = 40
But notice that SQRT 1600 = SQRT (16)(100) = 4(10) = 40
Try this:
Take an odd perfect square, such as 9. Square the largest whole number that is less than half of it. ( For 9 this would be 4). If you add this square to the original number what kind of number do you get? Try it with other odd perfect squares...
In this case, 9 + 16 = 25... hmmm... what's 25???
Numbers such as 1, 4, 9, 16, 25, 36, 49... are called square numbers or PERFECT SQUARES.
One of two EQUAL factors of a square is called the square root of the number. To denote a square root of a number we use a radical sign (looks like a check mark with an extension) See our textbook page 157.
Although we use a radical sign to denote cube roots, fourth roots and more, without a small number on the radical sign, we have come to call that the square root.
SQRT = stands for square root, since this blog will not let me use the proper symbol) √ is the closest to the symbol
so the SQRT of 25 is 5. Actually 5 is the principal square root. Since 5 X 5 = 25
There is another root because
(-5)(-5) = 25 but in this class we are primarily interested in the principal square root or the positive square root.
Evaluate the following:
SQRT 36 + SQRT 64 = 6 + 8 = 14
SQRT 100 = 10
Is it true that SQRT 36 + SQRT 64 = SQRT 100? No
You cannot add square roots in that manner.
However look at the following:
Evaluate
SQRT 225 = 15
(SQRT 9)(SQRT 25)= (3)(5) = 15
so
SQRT 225 = (SQRT 9)(SQRT 25)
Also notice that the SQRT 1600 = 40
But notice that SQRT 1600 = SQRT (16)(100) = 4(10) = 40
Try this:
Take an odd perfect square, such as 9. Square the largest whole number that is less than half of it. ( For 9 this would be 4). If you add this square to the original number what kind of number do you get? Try it with other odd perfect squares...
In this case, 9 + 16 = 25... hmmm... what's 25???
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