Transcript Prime Factorization [of a number]

Review of Prime Factorization
MATH 017
Intermediate Algebra
S. Rook
Overview
• Not in the textbook
– Should be a review from Fundamentals
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Prime & Composite Numbers
Factor Trees
Finding All 2 Pair Factors of a Number
Prime Factorization
– Greatest Common Factor (GCF)
– Least Common Multiple (LCM)
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Prime & Composite Numbers
Prime & Composite Numbers
• Prime numbers – a natural number that is
divisible by ONLY itself and 1.
– First 10 prime numbers (memorize these!):
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
• Composite numbers – a natural number
that is divisible by AT LEAST ONE other
number besides itself and 1.
• By convention, 1 is neither prime NOR
composite.
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Factor Trees
Factor Trees
• Start with the original number.
– If this number is prime, stop.
– Otherwise, look for a PRIME factor that evenly
divides the number
• Draw two branches from the number.
– On the left branch, write the prime number
– On the right branch, write the quotient of the original
number and the prime number.
• Keep repeating until both branches yield
prime numbers.
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Factor Trees (Example)
Ex 1: Find the factor tree for 36.
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Factor Trees (Example 1
Continued)
• Take note of 3 things from this example:
– If the left branch of the factor tree contains the prime
number, then only the right branch need be extended
downwards.
– The prime numbers were marked red to differentiate
them from composite numbers.
• The prime numbers are often circled when drawing factor
trees by hand.
– Instead of starting with 2, 3 could have been used.
• The result is the same: there are 2 twos and 2 threes.
• Different factor trees may exist for the same number, but
each will yield the same prime factors in the end.
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Factor Trees (Example)
Ex 2: Find the factor tree for 294.
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Finding All 2 Pair Factors of a
Number
Finding All Factors of a Number
• A very methodical process once a factor tree is
obtained.
• Start with 1 times the number.
• Any other factors must be between 1 and the
number.
• Use the circled prime numbers in the factor tree
to find subsequent factors.
• Repeat until you have exhausted all possibilities
in the gap.
• What results is ALL of the 2 pair factors!
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Finding All 2 Pair Factors of a
Number (Example)
Ex 3: Find all 2 pair factors of 36.
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Finding All 2 Pair Factors of a
Number (Example 3 continued)
• To help visualize the gap and to organize the
factors, use a tabular format. For example, 36:
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36
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18
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12
4
9
6
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• We can also gather the factors of 36:
1, 2, 3, 4, 6, 9, 12, 18, 36
• Listing ALL 2 pair factors of a number will be of
use when we cover factoring polynomials later.
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Finding All 2 Pair Factors of a
Number (Example)
Ex 4: Find all 2 pair factors of 294.
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Prime Factorization
Prime Factorization
• Prime Factorization [of a number]: a
product of primes that is equivalent to the
number.
• Usually written in exponential notation.
– E.g. 3 * 3 = 32
• Simple once the factor tree is obtained.
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Prime Factorization (Example)
Ex 5: Find the prime factorization of 36.
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Prime Factorization (Example)
Ex 6: Find the prime factorization of 294.
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Prime Factorization – GCF & LCM
• Greatest Common Factor (GCF) [of a group of
numbers]: the largest number that divides evenly into
all members of the group.
• Least Common Multiple (LCM) [of a group of
numbers]: the smallest number that all members of the
group divide evenly into.
• Do NOT mix these up!
– The GCF divides evenly into each member of the group
– Each member of the group divides evenly into the LCM.
• Easy to find either the GCF or LCM once the prime
factorization of each member in the group is obtained.
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Prime Factorization – GCF & LCM
(Example)
Ex 7: Find the GCF of 36 and 294.
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Prime Factorization – GCF & LCM
(Example)
Ex 8: Find the LCM of 36 and 294.
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Summary
• After studying these slides, you should know
how to do the following:
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Identify Prime & Composite Numbers
Construct Factor Trees
Find All 2 Pair Factors of a Number
Find the Prime Factorization and use it to:
• Identify the Greatest Common Factor (GCF)
• Identify the Least Common Multiple (LCM)
• Additional Practice
– Complete the online worksheet
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