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Review of Prime Factorization MATH 017 Intermediate Algebra S. Rook 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) 2 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. 4 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. 6 Factor Trees (Example) Ex 1: Find the factor tree for 36. 7 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. 8 Factor Trees (Example) Ex 2: Find the factor tree for 294. 9 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! 11 Finding All 2 Pair Factors of a Number (Example) Ex 3: Find all 2 pair factors of 36. 12 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 36 2 18 3 12 4 9 6 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. 13 Finding All 2 Pair Factors of a Number (Example) Ex 4: Find all 2 pair factors of 294. 14 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. 16 Prime Factorization (Example) Ex 5: Find the prime factorization of 36. 17 Prime Factorization (Example) Ex 6: Find the prime factorization of 294. 18 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. 19 Prime Factorization – GCF & LCM (Example) Ex 7: Find the GCF of 36 and 294. 20 Prime Factorization – GCF & LCM (Example) Ex 8: Find the LCM of 36 and 294. 21 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 22