Transcript MS133 - Mathematical, Computing, & Information Sciences

MS133
Chapter 4
Number Theory
Homework Questions
Test – Chapter 3
http://mcis.jsu.edu/faculty/mjohnson/ms133r3.html
Number Theory
• Triskaidekaphobia
• Palindromes
5678765
MOM
HANNAH
10/02/2001
46800864
BOB
RACE CAR
Number Theory
• Palindromes
When was the last time the date was a
palindrome before 10/02/2001?
08/31/1380
Number Theory
• Palindromes
When will the next date be a palindrome?
01/02/2010
Perfect Numbers
• 6 is a perfect number.
6=1x6
6=2x3
6=1+2+3
Perfect Numbers
• 28 is a perfect number.
28 = 1 x 28
28 = 2 x 14
28 = 4 x 7
28 = 1 + 2 + 4 + 7 + 14
Perfect Numbers
The next perfect number is 496
6, 28, 496, 8128, . . .
Pythagoreans
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•
•
•
•
Even – Male
Odd – Female
Marriage – 5
Pythagorean Theorem
Irrational Numbers
Fermat’s Last Theorem
• 1637
• an + bn = cn has no solution when n is a
natural number greater than 2.
• Complete proof in June 1993. (Over 350
years later)
• British mathematician at Princeton
University – Dr. Andrew Wiles (page 261)
• Let a, b and c be counting numbers such
that ab = c. a and b are said to be factors
or divisors of c.
• c is a multiple of a and c is a multiple of b.
• notation: a | c and b | c
(Read “a divides c” and “b divides c”)
• 4 x 7 = 28
factor x factor = product
• 4 and 7 are factors of 28 because they are
used in multiplying to get a product of 28.
• 28 ÷ 4 = 7
dividend ÷ divisor = quotient
• 4 and 7 are divisors of 28 because they
divide into 28 evenly.
• 28 is a multiple of 4 because it is the result
of “multiplying” a counting number by 4.
• 28 is a multiple of 7 because it is the result
of “multiplying” a counting number by 7.
Divisibility Rule for 2
List the multiples of 2 (those divisible by 2)
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, . . .
A number is divisible by 2 (an even
number) if the digit in the one’s place is 0,
2, 4, 6, or 8.
Example: 2 | 387,531,089,358 because
the digit in the one’s place is 8.
Divisibility Rule for 5
List the multiples of 5 (those divisible by 5)
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, . . .
A number is divisible by 5 if the digit in the
one’s place is 0, or 5.
Example: 5 | 7,678,934,620 because the
digit in the one’s place is 0.
• A number is divisible by 4 if the number
formed by the last 2 digits is divisible by 4.
Example: 4 | 56,720 because 4 | 20
• A number is divisible by 8 if the number
formed by the last 3 digits is divisible by 8.
Example: 8 | 3,567,240 because 8 | 240
What is the pattern?
2, 4, 8, . . .
What is the pattern for the divisibility rules
for 2, 4, 8, . . . ?
What do you suppose the divisibility rule for
16 would be?
How can you tell if something is divisible
by 25 (52)? (think quarters)
Make a conjecture:
A number is divisible by 25 (52) if
A number is divisible by 25 (52) if the
number formed by the last 2 digits is
divisible by 25.
Make a conjecture for a divisibility rule for
125 (53).
Multiples of 3 (Numbers divisible by 3):
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, . . .
Can you tell if something is divisible by 3
by looking in the one’s place?
A number is divisible by 3 if the sum of the
digits is divisible by 3.
Example: 3|315 because 3+1+5=9 and 3|9
• A number is divisible by 3 if the sum of the
digits is divisible by 3.
Example: 3|315 because 3+1+5=9 and 3|9
• A number is divisible by 9 if the sum of the
digits is divisible by 9.
Example: 9|783 because 7+8+3=18 and
9|18
A number is divisible by 7 if the number
formed without the one’s place, minus
twice the one’s place, is divisible by 7.
Example:
Which of the following are divisible by 7?
511
42,975
300,846
A number is divisible by 11 if the sum of
every other digit, minus the sum of the rest
of the digits is divisible by 11.
Example:
Which of the following is divisible by 11?
2,419,455,280
219,950,480
Relatively Prime
Two numbers (or a group of numbers) are
said to be relatively prime if they have no
common factors other than 1. The only
thing that will divide into both of them is 1.
Examples: 20 and 21 are relatively prime.
45 and 28 are relatively prime.
• How do you know if a number is divisible
by 10?
• If the number is divisible by 10, what else
does it have to be divisible by?
• What is the divisibility rule for 2?
• What is the divisibility rule for 5?
If a number is divisible by 2 AND the number is
divisible by 5, what has to be in the one’s place?
(What is the intersection of the two rules?)
A number is divisible by 10 if it is divisible
by 2 and 5.
All other tests will be like the test for 10, a
combination of the tests we have already
stated.
How can we check to see if a number is
divisible by 6?
How can we check to see if a number is
divisible by 12?
Will we check for 2 and 6 or 3 and 4?
Test that on the number 18 which we know
is not divisible by 12.
The numbers you use in combination must
be RELATIVELY PRIME.
Use the divisibility rules to list all the
factors (divisors) of 600:
Factors of 600
1,600
2,300
3,200
Notice that 2 is between 1 and 3 and the factor
that goes with 2 is between the factor that goes
with 1 and the factor that goes with 3.
300 is the only factor between 200 and 600.
How will we know when we are finished?
Factors of 600
1,600
2,300
3,200
4,150
5,120
6,100
8,75
Factors of 600
1,600
2,300
3,200
4,150
5,120
6,100
8,75
10,60
Factors of 600
1,600
2,300
3,200
4,150
5,120
6,100
8,75
10,60
12,50
Factors of 600
1,600
2,300
3,200
4,150
5,120
6,100
8,75
10,60
12,50
15,40
Factors of 600
1,600
2,300
3,200
4,150
5,120
6,100
8,75
10,60
12,50
15,40
20,30
Factors of 600
1,600
2,300
3,200
4,150
5,120
6,100
8,75
10,60
12,50
15,40
20,30
24,25
• A number is prime if it has exactly 2
factors, 1 and itself. The only way you can
multiply and get the number is to multiply 1
times the number.
• A number is composite if it has more than
2 factors.
• 1 and 0 are neither prime nor composite.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
Sieve of Eratosthenes
4
5
6
7
8
14 15 16 17 18
24 25 26 27 28
34 35 36 37 38
44 45 46 47 48
54 55 56 57 58
64 65 66 67 68
74 75 76 77 78
84 85 86 87 88
94 95 96 97 98
9 10
19 20
29 30
39 40
49 50
59 60
69 70
79 80
89 90
99 100
Sieve of Eratosthenes
http://nlvm.usu.edu/en/nav/frames_asid_158_g_3_t_1.html?open=instructions
How do we know if a number is
prime?
If it is divisible by only 1 and itself.
• What numbers will you check to see if they go
into it?
You will check to see if any prime numbers go
into the number. There is no need to check any
composite numbers. Any one of those would
have a prime number as a factor – which you
already checked.
Example: If the number is not divisible by 2 it
would not be divisible by any other even
number.
How do we know if a number is
prime?
• If it is divisible by only 1 and itself.
• You will check to see if any prime numbers go
into the number.
• How many prime numbers will you have to
check?
• If the number is 4 or more, you will have to
check 2. If the number is 9 or more you will
have to check 3 also. 25 or more, check 5 also.
49 or more check 7 also. 121 or more check 11
also, etc.
How do we know if a number is
prime?
• If it is divisible by only 1 and itself.
• You will check to see if any prime numbers go
into the number.
• You will check all the prime numbers until the
square of the prime number is bigger than the
number in question. Do not check that one. It
will not go into the number in question.
• If you have not found a factor, the number is
prime.
Composite Number
• If, during that process, you find a factor,
you may stop, tell what the factor is and
state that the number is composite.
• Remember, all it takes is one factor
besides 1 and the number itself to make it
be composite.
Tell whether each of the following is prime
or composite and explain why.
151?
231?
197?
Write the prime factorization of 600.
List all the factors of 600:
1, 600
2, 300
3, 200
4, 150
5, 120
6, 100
8, 75
10, 60
12, 50
15, 40
20, 30
24,25
Write the prime factorization of 600:
2•2•2•3•5•5 = 23•3•52
Day 2
Test Questions – Chapter 3
Homework Questions
Page 242
Homework Questions
Page 252
An architect is designing a display room
for an art museum. One wall is to be
covered with large square marble tiles.
The architect wants to use the largest tiles
possible. If the wall is to be 12 feet high
and 42 feet long, how large can the tiles
be?
An architect is designing a display room
for an art museum. One wall is to be
covered with large square marble tiles.
The architect wants to use the largest tiles
possible. If the wall is to be 12 feet high
and 42 feet long, how large can the tiles
be?
Dimensions of the tile (both the same)
must be a factor of 12 feet and 42 feet.
Factors of 12:
1,12
2,6
3,4
Factors of 42:
1,42
2,21
3,14
6,7
Dimensions of the tile (both the same) must be a
factor of 12 feet and 42 feet.
Factors of 12:
1,12
2,6
3,4
Factors of 42:
1,42
2,21
3,14
6,7
He wanted the largest possible tile.
The tiles should be 6 ft by 6 ft.
The Greatest Common Factor (Divisor) (GCF or
GCD) of two or more numbers is the biggest
thing that will divide into both (or all) of the
numbers.
factor num bers
GCF num bers
Example:
Find the GCF of 18 and 45.
Factors of 18:
1, 18
2, 9
3, 6
GCF(18, 45) =
Factors of 45:
1, 45
3, 15
5, 9
Find the GCF(18,45) by prime factorization.
Find the GCF(18,45) by prime factorization.
2)18
3)9
3
3)45
3)15
5
2•3•3
3•3•5
GCF = 3•3 = 9
GCF(504, 3675)
GCF(504, 3675)
2)504
2)252
2)126
3)63
3)21
7
504 = 23•32•7
GCF = 3•7 = 21
3)3675
5)1225
5)245
7)49
7
3675 = 3•52•72
Find the GCF of 504 and 3675
)504
3675
Euclidean Algorithm for Finding the
GCF of 504 and 3675
504)3675
Euclidean Algorithm for Finding the
GCF of 504 and 3675
7
504)3675
3528
147
Euclidean Algorithm for Finding the
GCF of 504 and 3675
7
504)3675
3528
147
147)504
Euclidean Algorithm for Finding the
GCF of 504 and 3675
7
504)3675
3528
147
3
147)504
441
63
Euclidean Algorithm for Finding the
GCF of 504 and 3675
7
504)3675
3528
147
3
147)504
441
63
63)147
Euclidean Algorithm for Finding the
GCF of 504 and 3675
7
504)3675
3528
147
3
147)504
441
63
2
63)147
126
21
Euclidean Algorithm for Finding the
GCF of 504 and 3675
7
504)3675
3528
147
21)63
3
147)504
441
63
2
63)147
126
21
Euclidean Algorithm for Finding the
GCF of 504 and 3675
7
504)3675
3528
147
3
21)63
63
0
3
147)504
441
63
GCF = 21
2
63)147
126
21
GCF(18,411, 1649)
The Least Common Multiple (LCM) of two or
more numbers is the smallest number that
they will all divide into.
num bers m ultiples
num bers LCM
Find the LCM(9,15)
Multiples of 9:
Multiples of 15:
Find the LCM(9,15)
Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72,
81, 90, 99, 108, 117, 126, 135, 144, . . .
Multiples of 15: 15, 30, 45, 60, 75, 90, 105,
120, 135, 150, 165, 180, 195, 210, . . .
LCM(9,15) =
Find the LCM(9,15)
Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72,
81, 90, 99, 108, 117, 126, 135, 144, . . .
Multiples of 15: 15, 30, 45, 60, 75, 90, 105,
120, 135, 150, 165, 180, 195, 210, . . .
LCM(9,15) = 45
Find the LCM(9,15) by prime factorization.
Find the LCM(9,15) by prime factorization.
3)9
3
3)15
5
9 = 32
15 = 3•5
LCM = 32•5 = 45
GCF(504, 3675)
2)504
2)252
2)126
3)63
3)21
7
504 = 23•32•7
3)3675
5)1225
5)245
7)49
7
3675 = 3•52•72
GCF = 3•7 = 21
LCM = 23•32•52•72 = 88,200
__________
GCF ) NUMBERS
_____
NUMBERS ) LCM
r = 25•34•52•73•112
s = 24•32•53•75•13
GCF =
LCM =
Day 3
Homework Questions
Page 264
Factors and Multiples With Rods
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Common Length?
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Common Length = 8
GCF = 8
GCF(24,40) =
GCF(24,40) = GCF(16,24)
= GCF(8,16)
= GCF(8,8)
=8
• • •|
• • • • •|
• • •|• • •|
• • • • •|
• • •|• • •|
• • • • •|• • • • •|
• • •|• • •|• • •|
• • • • •|• • • • •|
• • •|• • •|• • •|• • •|
• • • • •|• • • • •|
• • •|• • •|• • •|• • •|
• • • • •|• • • • •|• • • • •
• • •|• • •|• • •|• • •|• • •
• • • • •|• • • • •|• • • • •
Common Length?
• • •|• • •|• • •|• • •|• • •
• • • • •|• • • • •|• • • • •
Common Length = 15
LCM = 15
• GCF → break into pieces to find a
common length
• LCM → multiply to get a common length
GCF and LCM with Venn Diagrams
GCF(24, 36)
LCM(24, 36)
24 = 23 • 3
36 = 22 • 32
GCF(24, 36)
LCM(24, 36)
24 = 23 • 3
36 = 22 • 32
36
24
2
2
2
3
3
GCF(24, 36)
LCM(24, 36)
24 = 23 • 3
36 = 22 • 32
36
24
2
2
2
3
GCF =
3
GCF(24, 36)
LCM(24, 36)
24 = 23 • 3
36 = 22 • 32
36
24
2
2
2
3
GCF = 2 • 2 • 3 = 12
3
GCF(24, 36)
LCM(24, 36)
24 = 23 • 3
36 = 22 • 32
36
24
2
2
2
3
GCF = 2 • 2 • 3 = 12
LCM =
3
GCF(24, 36)
LCM(24, 36)
24 = 23 • 3
36 = 22 • 32
36
24
2
2
2
3
GCF = 2 • 2 • 3 = 12
LCM =2 • 2 • 2 • 3 • 3 = 72
3