Transcript 3*5 - Edublogs

By Nanako
There are two different kinds of numbers. Prime numbers and composite
numbers.
Prime numbers: A number with exactly 2 factors. Not “a number divisible by only 1 and
itself” because one is not prime! It only has one factor!
Composite numbers: A number with more than 2 factors.
Since 60 has 12 factors, which is more than 2, it is a composite number.
1,2,3,4,5,6,10,12,15,20, 30, 60
These are all of 60’s factors, but 60’s proper factors are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30
60’s factor pairs are: 1 and 60, 2 and 30, 3 and 20, 4 and 15,
5 and 12, 6 and 10. Factor pairs are 2 numbers that are multiplied
x
=
together to equal a number. For example,
This means
I know that I haven’t missed any because I used a factor
that
and
rainbow to find the factor pairs. When you use a factor rainbow to
are factor
find factor pairs, it lets you find all the factors and factor pairs of a
pairs of
number without missing anything.
1
2
3
4
5
6
10
12
15
20
30
60
First of all, this is the definition for abundant, deficient, and perfect numbers.
Abundant: A number with proper factors that have a sum greater tan the number itself.
Example: 12 is abundant because it’s proper factors add up to 16, which is more than
12…1+2+3+4+6=16
16>12
Deficient: A number with proper factors that have a sum less than the number itself.
Example: 15 is deficient because it’s proper factors add up to 9, which is less than
15…1+3+5=9
9<15
Perfect: A number with proper factors that have a sum equal to the number itself.
Example: 28 is perfect because it’s proper factors add up to 28, which is exactly the
number itself (28). 1+2+4+7+14= 28
28=28
If you add up all of 60’s proper factors, then you get 168.
1+2+3+4+5+6+10+12+15+20+30 = 168.
168>60
This means that 60 is an abundant number.
60 is an even number, not an odd number. I know this because even
numbers always end in 0,2,4,6,8, and odd numbers end in 1,3,5,7,9. 60 ends in
a zero, 60, which means that it is an even number.
Every number is either even of odd. After an even number is an odd
number, then it is an even number again. Numbers go in a pattern: Even, odd,
even, odd, even, odd…Think of it like this.
Even
odd
even
odd
even
odd
even
odd
even
The prime factorization of 60 is
2*2*3*5 or 22*31*51. 22*31*51 is the same as
2 x 2 x 3 x 5, but it is just written in index
notation.
60
2
* 30
3
The prime factorization of a number
is the prime numbers that are multiplied
together to equal the number itself.
* 10
5
*
2
Prime factorization of 60:
2*2*3*5 or 22*31*51
To find the prime factorization of a number,
you need to make a factor tree. A factor tree
helps you find the prime factorization of a
number, because you are breaking down all of
the factors until they are all prime. As the
prime numbers come up, you circle them so
you will know what the factorization is.
Multiples of numbers go on forever…so these are just the
first 12 multiples of 60.
60,120,180,240,300,360,420,480,540,600,660,720
60 x any number = a multiple of 60.
For example, 60*2 = 120, so 120 is a multiple of 60.
60*3 = 180, so 180 is a multiple of 60.
A common factor is a factor that 2 or more numbers have in common. For
example, when you line up the factors of 60 and 15, you will notice that they have 1,3,5,
and 15 in common. Those are the common factors of 60 and 15.
60’s factors: 1,2,3,4,5,6,10,12,15,20,30,60
15’s factors: 1,3,5,15
A common multiple is a multiple that 2 or more numbers have in common. For
example, when you line up the multiples (1-600) of 60 and 30, you will notice that they
have 60,120,180,240,300,360,420,480,540,and 600 in common. Those are just some of
the common multiples of 60 and 30.
60’s multiples (1-600):
60,120,180,240,300,360,420,480,540,600
30’s multiples (1-600):
30,60,90,120,150,180,210,240,270,300,330,360,390,420,450,480,510,540,570,600
60
60
15
20 *
2*2
3*5
5
15
3
3
* 4
GCF =3*5
=15
2
*
2
*
5
60’s prime
factorization:
2*2*3*5
15’s prime
factorization:
3*5
Common factors:
3*5
LCM =2*2*3*5
=4*15
=60
One way to find the GCF and LCM of 60 and 15 is to use a Venn diagram. First, though,
you need to find the prime factorizations of both numbers. When you do that, you find out that the
prime factorization of 60 is 2*2*3*5, and that the prime factorization of 15 is 3*5. Then, line up the
prime factorizations of both numbers, and find the common factors. Using that information, make a
Venn diagram in which one circle is for 60’s prime factorization, one for 15’s prime factorization, and
the circle in the middle for the common factors. In the middle circle put in the common factors (3*5).
In 60, only write the 2*2, because the 3*5 is already written in the middle. Since 15 doesn’t have
any other factors other than 2*2, just leave it as it is. Lastly, for the GCF and LCM. For the GCF,
multiply the numbers in the middle, which is 3*5, so the GCF is 15. For the LCM, you need to
multiply all of the numbers in the diagram, so you do 2*2*3*5, which equals 60. The LCM is 60.
60
3
5
15
3
5
GCF
3
5
3*5=15
3
5
2*2*3*5 = 60
LCM
2
2
2
GCF = 15
LCM = 60
2
Another way to find the GCF and LCM of 60 and 15 is to make a table like the one
above. In the first 2 columns, write 60 and 15, which is the 2 numbers that you are finding the GCF
and LCM of. In the 3rd and 4th columns, write GCF and LCM. Now, starting with 60, write 60’s prime
factorization (2*2*3*5) in the next 4 columns. Next, we are going to write the prime factorization of
15 (3*5) in the next row, but since 3*5 is already written once in 60’s prime factorization, just write it
underneath it. Now, to find the GCF, take the factors that 60 and 15 have in common (a factor that
is in both number’s prime factorization) and multiply those numbers. So you would do 3*5 which
equals 15. The GCF of 60 and 15 is 15. Lastly, for the LCM, multiply all of the factors of both 60 and
15. Make sure you only multiply the common factors (3*5) once. So you would do 2*2*3*5 which
equals 60, so the LCM of 60 and 15 is 60.
60
15
60
3*5
2
15
3
* 30
2
* 15
2*7
3
2*7
*
5
*
56
5
Prime factorization of:
60: 2*2*3*5
15: 3*5
56: 2*2*2*7
7
*
8
4 * 2
2
*
2
56
To find the GCF and LCM of 60, 15, and 6 (3 numbers), first, you need to find the prime
factorization of all the numbers. 60 = 2*2*3*5, 15 = 3*5, and 56 = 2*2*2*7. Next, you need to find
what factors they have in common. For example, 60 and 56 have 2*2 in common, and 60 and 15
have 3*5 in common. Now, you will need to put the numbers into the Venn diagram. The numbers
that 60 and 56 have in common go in the circle in between the circle for 60 and 56, and the circle
between 60 an d 15 is the circle for the factors that they have in common, etc. Once you have put
the numbers in, to find the GCF, you multiply the numbers in the very middle of the diagram, but in
this case, there is nothing in the middle, which means that the GCF is 1. Lastly, the LCM is all of the
numbers in the diagram, so 2*2*3*5*7*7 = 2940. The GCF is 1, and the LCM is 2940.
300
3
Prime factorization of:
300:2*2*3*5*5
360:2*2*2*3*3*5
* 100
2
300
360
* 50
5
* 10
5
5
2*2*3*5
* 2
360
6 * 60
GCF = 2*2*3*5
= 4*15
= 60
2 * 3 * 2 * 30
3
* 10
5
*
2
2*3
Picture URL’S:
http://www.colchesterhospital.nhs.uk/images/NHS%2060%20mosaic.
jpg
http://rwridley.files.wordpress.com/2009/09/60.png
http://www.steinardanielsen.com/wpcontent/uploads/2010/08/hello_my-urlis.jpg
http://math.phillipmartin.info/math_prime_composite.gif