Download A number like 24 that has factors other than 1 and

AN INTRODUCTORY ACTIVITY
You have 24 square tiles that you want to
arrange in the shape of a rectangle (in
such a way that the rectangle is
completely filled with tiles).
CHAPTER 4
NUMBER THEORY
|
Describe all the possible arrangements.
|
What is the significance of each of these
arrangements?
Section 4.1
Factors and Divisibility
FACTORS
Each arrangement of tiles gives you
information about the factors of 24.
A FURTHER INVESTIGATION
|
A number A is a _____________of a number N:
You can only make one rectangle, a 1 by 13.
You cannot divide 13 evenly into groups of
any
y size other than 1 or 13. That is,, 13 has
no factors other than 1 or 13.
if you can divide N evenly into groups of size A
( if you can fi
(or
find
d a whole
h l number
b B such
h th
thatt N
= B x A)
A number that has only 1 and itself
as factors is a ________________.
A number like 24 that has factors
other than 1 and itself is called a
______________________.
OR
if you can divide N into A groups all of the same
size (i.e. can find a whole number B such that N =
A x B).
A FURTHER INVESTIGATION (CONTINUED)
A FURTHER INVESTIGATION (CONTINUED)
|
|
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If you have more than 24 tiles, will you
necessarily be able to make more rectangles
than you could in the first problem? Try
some experiments.
Not necessarily.
necessarily For example
example, if you had 25 tiles
tiles,
you could only make 2 different rectangles;
1 by 25 and 5 by 5.
What if there are only 13 tiles? Now how
many rectangles can you make?
|
|
John has lots of 1 inch by 1 inch square tiles.
John will use his tiles to make rectangles (in such
a way that each rectangle is filled with tiles), each
of which has one side that is 4 inches long.
What are the areas of the rectangles that John can
make? (The area of a rectangle will be the number
of tiles composing the rectangle.)
Is the connection of this problem with factors or
multiples?
John would be creating _________________ of 4.
A 4 by 1 rectangle would have 4 tiles; a 4 by 2
rectangle would have 8 tiles.
(2 groups of 4 = 2x4 = 8), etc.
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NUMBER THEORY
MULTIPLES
|
|
For a whole number A, a multiple of A is a
number N such that N = B x A or N = A x B
where B is a whole number.
The number A can be thought of as the
number of groups or the size of a group and
the _________________________________________.
FACTOR TEST
To find all the factors of a number n, test only
those natural numbers that are no greater
than the square root of the number, √⎯n.
| Examples: Find the factors of the following
numbers using your calculator.
calculator
1) 78
2) 156
3) 252
|
|
Number theory is the study of the characteristics
of and relationships involving the natural
numbers.
Many
y of the useful characterizations of the
natural numbers are based on information about
the ________________ and ________________ of a
number.
DIVISIBILITY TESTS
|
Divisibility by 2
0, 2, 4, 6, 8, …, 28, 30, 32, 34, …
(A natural number n is divisible by 2 iff
______________________________________________
______________________________________________
|
|
Divisibility by 5
0, 5, 10, 15, …, 25, 30, 35, …
(A natural number n is divisible by 5 iff
______________________________________________
|
Divisibility by 10
0, 10, 20, …, 100, 110, 120, …
(A natural number n is divisible by 10 iff
_____________________________________________
DIVISIBILITY TESTS (CONTINUED)
|
Divisibility by 3
0, 3, 6, …, 12, 15, 18, …
(A natural number n is divisible by 3 iff
______________________________________________
|
Divisibility by 9
0, 9, 18, 27, …, 54, 63, 72, …
(A natural number n is divisible by 9 iff
______________________________________________
|
Divisibility by 6
A natural number n is divisible by 6 iff
______________________________________________
|
Divisibility by 4
A natural number n is divisible by 4 iff
______________________________________________
______________________________________________
DIVISIBILITY TESTS (CONTINUED)
FYI (FUN FACTS)
| Divisibility
by 8
A natural number n is divisible by 8 iff the
number represented by its last three digits is
divisible by 8.
| Divisibility
by 7
A natural number n is divisible by 7 iff the
number formed by subtracting twice the last digit
from the number formed by all the digits but the
last is divisible by 7.
| Divisibility
by 11
A natural number n is divisible by 11 iff the sum
of the digits in the even-powered places minus the
sum of the digits in the odd-powered places is
divisible by 11.
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EXAMPLES
a) Fill in the blanks so that the number is divisible
by 2, but not 5 or 10.
8 6 3 , ___ ___ ___
b) Complete
Co p ete the
t e number
u be so itt iss divisible
d v s b e by 3 and
a d 9.
1 0 , 8 2 1 , 7 ___ ___
TO
USING FACTORS
CLASSIFY NATURAL NUMBERS
| Even
| Odd
numbers: have 2 as a factor
numbers: do not have 2 as a factor
| Squares:
have an odd number of factors
(one pair of factors is made up of the
same number)
25: 1, 5, 25
USING FACTORS TO CLASSIFY NATURAL NUMBERS
Perfect numbers: sum of their factors that are
less than the number (its _________________)
equals the number
| Deficient numbers: sum of their proper factors
is less than the number
| Abundant
Ab d
numbers
b
: sum off th
their
i proper
factors is greater than the number
| Amicable numbers: two numbers such that the
sum of the proper factors of the first
number equals the second number and the
sum of the proper factors of the second
number equals the first number
|
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