Download Aim 4 - Prime Factorization Classwork

Aim 4 ­ Prime Factorization Classwork.docx92914.notebook
September 28, 2015
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Aim 4: Prime Factorization
Do Now
A prime number is a whole number greater than 1 with only two factors, 1 and itself.
A composite number is a whole number greater than 1 with more than two factors.
0 and 1 are neither !!!
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Prime factorization is writing a number as a product of only prime numbers.
1. Write the prime factorization of 60.
Method: Factor Tree Factoring until we are left with only prime factors.
http://en.wikipedia.org/wiki/Divisibility_rule
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2. Write the prime factorization of 126.
3. Write the prime factorization of 225.
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True or False? Why?
4. A number can have more than one factor tree. _______
5. A number can have more than one prime factorization. ______
6. Two different numbers can have same prime factorization. ______
1) True.
2) False. Only one prime factorization for any given composite #.
3) False. The same factors will always have the same product.
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7) What is the difference between a factor and a multiple?
{Factors of 30} {Multiples of 30}
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The greatest common factor (GCF) of 2 or more whole numbers is the greatest whole number that divides evenly into each number.
8) Find the GCF of 28 and 35
Method 1: List out all the factors Method 2: Prime Factorization
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9) Find the GCF of 24, 36, and 42.
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10) Find the GCF of 18, 27, and 36.
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The least common multiple (LCM) of 2 or more numbers is the smallest number that can be divided evenly by both(all) numbers. 11) Find the LCM of 15 and 18
Method 1: List out multiples until you find one in common
Method 2: Prime Factorization
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12) Find the LCM of 10, 12 and 15
13) Find the GCF and the LCM of 24, 36 and 48.
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14. The width of a rectangular swimming pool (in feet) is a prime number greater than 10. The area of the pool cover is 408 square feet. What are the length and width of the pool cover?
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Aim 6: Least Common Multiple (LCM)
Do Now
Find the GCF
1) 99, 63
2) 195, 171
3) 345, 180
4) 378, 270
5)14X2 , 38X3
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Can you find the LCM (5, 6, 10) by
3 methods we learned in class?
LCM = GCF x leftovers
choose the highest power
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Find the GCF and LCM of 84, 56, and 66 by using the three methods.
What do you think about
each method?
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List all the Factors
{ Factors of 84} = {1, 2, 3, 4, 6, 7, 12, 14 , 21, 28, 42, 84}
{ Factors of 56} = { 1, 2, 4, 7, 8, 14, 28, 56}
{ Factors of 66} = { 1, 2, 3, 6, 11, 22, 33, 66}
1, 2 are the common factors
2 is the greatest factor
GCF = 2
{ Multiples of 84} = { 84, 168, 252, 336, 420, 504, 588, 672, 756, 840, 924, 1008, 1092, 1176, 1260, 1344, 1428, 1512, 1596, 1680, 1764, 1848, 1932,....}
{ Multiples of 56} = { 56, 112, 168, 224, 280, 336, 392, 448, 504, 560, 616, 672, 728, 784, 840, 896, 952, 1008, 1064, 1120, 1176, 1232, 1288, 1344, 1400, 1456, 1512, 1568, 1624, 1680, 1737, 1792, 1848, 1904,....}
{ Multiples of 66} = { 66, 132, 198, 264, 330, 396, 462, 528, 594, 660, 726, 792, 858, 924, 990, 1056, 1122, 1188, 1254, 1320, 1386, 1452, 1518, 1584, 1650, 1716, 1782, 1848, 1914,...}
1848 is the least value in all lists
LCM = 1848
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Chinese Ladder Method
Prime Factorization
2 I___84_____56____66__
42 28 33
84 = 22 x 3 x 7
56 = 23 x 7 66 = 2 x 3 x 11
GCF = 2
LCM = 2 x 42 x 28 x 33 = 77616
GCF = 2
LCM = 23 x 3 x 7 x 11 = 1,848
1848 / 84 = 22
1848 / 56 = 33
1848 / 66 = 28
GCF (22, 33, 28) = 1
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Find the LCM by using prime factorization
1) 4, 7
2) 4, 12
3) 3, 6, 14
4) 6, 12, 16
5) 25, 110, 45
6) 9, 15, 27
7) 1, 9, 27, 36 8) 8, 12, 18, 20
1) 28
2) 12
3) 42
4) 48
5) 4950
6) 135
7) 108
8) 360
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1) Describe a way to remember the difference between GCF and LCM.
2) Can two numbers have the same LCM as their GCF?
3) List common multiples of 6 and 9 that are not the LCM.
1) GCF is a factor and will be equal to or smaller than each number.
LCM is a multiple of each number and must be equal to or greater than each number. 2) Only if the two numbers are the same.
3) LCM (6, 9) = 18 {36, 54, 72, 90,...}
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Find the prime factorization
1) 342
2) 765
GCF or LCM? Show work.
3) Jack is cutting a 72­inch­long board and 54­inch­long board to make shelves. He wants the shelves to be the same length while not wasting any wood. What is the longest possible length of the shelves?
4) In the United States, a president is elected every four years. Members of the House of Representatives are elected every two years. Senators are elected every six years. If a voter had the opportunity to vote for a president, a representative, and a senator in 1996, what will be the next year the voter has a chance to make a choice for a president, a representative, and the same Senate set?
Complete your answer by drawing a Venn diagram. 5) 40 people in survey. 20 people play hockey. 27 people play soccer. 5 people play neither. a) How many people play both? b) How many people play only soccer?
c) How many people play only hockey?
6) The student council conducted a poll asking the students to choose which of three candies they enjoyed eating: chocolate, jelly beans or lollipops. These were the results: 200 liked chocolate 150 liked jelly beans 100 liked lollipops
20 liked chocolate and jelly beans only 20 liked chocolate and lollipops only 10 liked jelly beans and lollipops only 20 liked all three candies
How many people were polled? 7) 24 students like baseball only 8 students like football only 6 students like baseball and soccer 4 students like baseball and football 2 students like football and soccer
2 students like all three sports 66 students participated in the survey
How many students chose only soccer?
Simplify
8)
9)
10)
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Aim
Scientific Notation
Do Now
Find the LCM
1) 39, 26
2) 49, 56
3) 17, 51, 8
4) 10, 100, 1000
5) 294W, 308
1) 78
2) 392
3) 408
4) 1000
5) 6468W
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Complete the table and look for a pattern.
Power of 10
Factors
Product
10 1
10 10
102
10 x 10 100
103
104
106
109 1010
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Scientific Notation
writing a number between 1 and 10 times 10 to a power
1790000
Standard Form 6
10
1.79
Scientific Notation x greater than or equal to 1
and less than 10
a power of 10
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Write in Standard Form
5. 34 x 10 4
Multiply by a positive power of 10 moves the decimal point to the right.
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0.17
Standard Form
Scientific Notation
1.7 10­1
x
greater than or equal to 1
and less than 10
a power of 10
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Write in Standard Form
3.27 x 10 ­3
Multiply by a negative power of 10 moves the decimal point to the left.
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Write in Scientific Notation
Write in Standard Form
1) 150
2) 34500
3) 164000
4) 17770
5) 260.7
6) 14.156
1) 7.42 x 10 5
2) 1.32 x 10 4
3) 3.714 x 10 5
4) 2.081 x 10 3
5) 9.931 x 10 7
Write in Scientific Notation
Write in Standard Form
1) 0.0078
2) 0.456
3) 0.00806
4) 0.012098
5) 0.00009753
1) 6.1 x 10­2
2) 1.1 x 10­3
3) 8.62 x 10­5
4) 2.45 x 10­4
5) 6.299 x 10­6
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GCF or LCM? Show work.
1) Jack is cutting a 72­inch­long board and 54­inch­long board to make shelves. He wants the shelves to be the same length while not wasting any wood. What is the longest possible length of the shelves?
2) In the United States, a president is elected every four years. Members of the House of Representatives are elected every two years. Senators are elected every six years. If a voter had the opportunity to vote for a president, a representative, and a senator in 1996, what will be the next year the voter has a chance to make a choice for a president, a representative, and the same Senate set?
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Write in Scientific Notation
3) The distance from Milford to Loveland is 326 kilometers. If there are 1,000 meters in a kilometer, use scientific notation to write the distance from Milford to Loveland in meters.
4) The number of visitors from various countries to the United States in a recent year are list in the table. Order the countries according to the number of visitors from greatest to least. Country
# of vistors
Canada
1.46 x 107
France
1.1 x 107
Germany
1.8 x 108
Japan
5.1 x 106
Mexico
1.03 x 107
United Kingdom 4.7 x 105
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1) Order from the least to the greatest
3.9 x 10 6 3800000 4.2 x 10
5
2) Is 12.5 x 10 7 written in scientific notation? Why?
3) True or False
a) 6.5 x 10 3 > 6.4 x 10 4
b) 2.1 x 10 ­2 > 2.2 x 10 ­3
1) 420,000 3,800,000 3,900,000
2) No, 1.25 x 108
3) a) False, 6500 < 640000
b) True, 0.021 > 0.0022
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Find the GCF & LCM by using Prime Factorization.
1) 2420, 1485
2420 = 2 2 x 5 x 112
1485 = 3 3 x 5 x 11
GCF = 55
LCM = 2 2 x 33 x 5 x 112 = 65,340
2) 2904, 2970
2904 = 2 3 x 3 x 112
2970 = 2 x 3 3 x 5 x 11
GCF = 66
LCM = 2 3 x 33 x 5 x 112 = 130,680
3) 50, 55, 40
50 = 2 x 5 2
55 = 5 x 11
40 = 2 3 x 5
GCF = 5
LCM = 2 3 x 52 x 11 =2200
4) 4095, 2394, 882
4095 = 3 2 x 5 x 7 x 13
2394 = 2 x 3 2 x 7 x 19
882 = 2 x 3 2 x 72
GCF = 63
LCM = 2 x 3 2 x 5 x 7 2 x 13 x 19 = 1,089,270
5) 609, 441, 540
609 = 3 x 7 x 29
441 = 3 2 x 72
540 = 2 2 x 33 x 5 GCF = 3
LCM = 2 2 x 33 x 5 x 7 2 x 29 = 767,340
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Prime Factorization Worksheet Name
Date
Tell whether each number is prime, composite, or neither.
1) 1 ________________
2) 45 ________________
3) 87 ________________
4) 15 ________________
5) 53 ________________
6) 110 ________________
7) 29 ________________
8) 56 ________________
9) 93 ________________
10) 44 ________________
11) 31 ________________
12) 114 _______________
Find the prime factorization of each number.
13) 675
14) 1,375
15) 18,522
16) 13,720
17) 21, 505
18) 154,350
19) 2,431,000
20) 517,810,293
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Aim 5: Greatest Common Factor (GCF)
Do Now
Evaluate.
1) (5 + 3 x 20) ÷13 + 32
2) (3 x 3 ­3)2 ÷3 + 3
3) (3 ÷ 3) + 3 x (3 3 ­3)
Complete the equation.
4) 32 __ 6 ( 0 ) 2 __ 8 = 40
5) 8 0 __ 3 √ 25 __ 4 (6) __ 3 = 8
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1. Find the GCF of the numbers 24, 60 and 36. Method 1: List the factors of each number.
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Method 2: Use the prime factorization for each number.
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Find the GCF
1) 22, 121
2) 60, 126
3) 18, 27, 36
4) 28, 42, 210
5) 30, 45, 60, 105
6) 13, 39, 52, 78
7) 2, 3, 4, 5, 7
1) 11
2) 6
3) 9
4) 14
5) 15
6) 13
7) 1
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1) What pair of numbers has a GCF that is a prime number, 48 and 90 or 105 and 56?
2) Is every factor of the GCF of two numbers also a factor of each number?
1) GCF (105, 56) is 7
2) Yes. Every factor of the GCF is also a factor of each number. 59
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