Transcript prime factor

Factors and Multiples
What is a Prime Number?
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A prime number has exactly two different divisors - one
and itself.
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Is one a prime number?
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Zero?
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Why or why not?
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A composite number is a number with 3 or more
factors.
Prime Numbers
• “Prime
numbers are what is left when you
have taken all the patterns away. I think
prime numbers are like life. They are very
logical but you could never work out the
rules, even if you spent all your time
thinking about them.”
• Mark
Haddon, The Curious Incident of
the Dog in the Night-time
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I Claim...
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The numbers left are prime numbers.
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True or False, and why?
Prime Factors - Key Ideas
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When a divisor of a number is a prime number, it is
called a prime factor.
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The prime factorization of a natural number is the
number written as a product of its prime factors.
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Every composite number can be expressed uniquely as
a product of prime factors.
Fun Fact
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A perfect number is one whose proper divisors sum to the
number itself.
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e.g. The number 6 has proper divisors 1, 2 and 3
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1+2+3=6
28 has divisors 1, 2, 4, 7 and 14
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1 + 2 + 4 + 7 + 14 = 28.
Factor tree
1980
198
22
11
10
9
2
3
5
3
1980  2  2  3 3 5 11
 2  3  5 11
3
2
2
Division by Prime Factors
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1980 ÷ 2 = 990
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990 ÷ 2 = 495
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495 ÷ 3 = 165
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165 ÷ 3 = 55
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55 ÷ 5 = 11
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11 ÷ 11 = 1
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The prime factors of 1980 are 2, 3, 5 and 11
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The prime factorization of 1980 is (2)(2)(3)(3)(5)(11) = (22)(32)(5)(11)
What letters do these words
have in common?
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sofa and mast
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scissors and stick
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sic
kitten and knit
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as
knit
Canada and banana
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anaa
What if...
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We wrote Canada and banana as algebraic terms?
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Canada = a3Cdn
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banana = a3bn
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The letters they have in common are aaan, or a3n
What about scissors and stick?
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scissors = ciors4
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stick = cikst
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The letters they have in common are cis (notice they only have one factor of s in
common, so we can only take one, not four)
Greatest Common Factor
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The greatest common factor of two or more numbers is
the greatest divisor the numbers have in common.
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e.g. The factors of 8 are 1, 2, 4 and 8.
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The factors of 12 are 1, 2, 3, 4, 6 and 12.
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The greatest common factor of 8 and 12 is 4.
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We write this as GCF(8, 12) = 4.
List Method
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List all the factors of each number.
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Find all the factors common to all numbers.
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The largest factor is the GCF.
List Method
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Find the GCF of 12 and 45.
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Find GCF(18, 54)
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Find GCF(16, 48, 80)
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What are the advantages of this method?
Disadvantages?
Try This...
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Find a partner. How many push-ups can you do in one
minute? Time each other.
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Find the prime factorization of your own number.
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What do you have in common? Use the lowest power
of each prime.
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Multiply the common primes together. Is this number a
factor of both numbers?
Prime Factorization Method
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Find the prime factorization of each number
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Find the lowest power of each factor
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Multiply the lowest power of each factor together
Prime Factorization Method
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12 = (22)(3)
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45 = (32)(5)
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No common factor of 2
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Lowest power of the common factor of 3 = 1
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No common factor of 5
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GCF (12, 45) = 3
Prime Factorization Method
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GCF (28, 98)
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GCF (32, 80)
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GCF (27, 36,126)
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What are the advantages and disadvantages to this
method?
Example GCF Word Problem
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A chocolate company has developed a new chocolate
bar. They have decided that it should be 110 mm by
275 mm, and want to divide it evenly into the largest
squares possible. How big can the squares be?
Une question pour vous
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I’m knitting a sweater that
has two different
patterns. One pattern
repeats every 12 rows,
and the other every 20.
What is the minimum
number of rows the
sweater needs to be for
both patterns to be
finished at the end?
Lowest Common Multiple
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The least common multiple of two or more numbers is
the least number that is divisible by each number
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e.g. Some multiples of 12 are 12, 24, 36, 48, 60, 72,
84, 96, 108
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Some multiples of 20 are 20, 40, 60, 80, 100, 120
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The least common multiple is 60
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We write this as LCM (12, 20) = 60
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Lowest common denominator, Elimination
Multiple List Method
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List multiples each number
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How far do you need to list?
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Find the lowest multiple common to all
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Find LCM (6, 22)
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LCM (18, 20, 30)
Advantages? Disadvantages?
Prime Factorization Method
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WAIT!!! We did a prime factorization method for GCF!!!
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How did we find GCF with prime factorization?
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LCM is the opposite
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Instead of finding taking the smallest powers, we
take the largest
Prime Factorization Method
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Find LCM (6, 22)
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24 = 23 3
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54 = 2 33
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LCM (6, 22) = 23 33 = 216
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Advantages? Disadvantages?
GCF vs. LCM
GCF
Compare - How are
they the same?
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How are they different?
How are they the same?
Contrast - How are they
different?
How is each one used?
Examples
LCM