Review of Prime Factorization

MATH 017

Intermediate Algebra

S. Rook

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Overview

• Not in the textbook– Should be a review from Fundamentals

• Prime & Composite Numbers

• Factor Trees

• Finding All 2 Pair Factors of a Number

• Prime Factorization– Greatest Common Factor (GCF)– Least Common Multiple (LCM)

Prime & Composite Numbers

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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.

Factor Trees

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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.

Finding All 2 Pair Factors of a Number

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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:

1 362 183 124 96 6

• 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.

Prime Factorization

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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:– 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