Section 5 2 Greatest Common Factors and Lowest

Greatest Common Factor (GCF) Definition: The GCF of two whole numbers a and b

Greatest Common Factor (GCF) Methods for finding GCF(a, b) (method I is straight forward

One more exercise Find the GCF(882, 924) (click for answer. ) Answer: 882 =

(method II is still difficult to use if the numbers are too large and

(III) Euclidean Algorithm – depends on reducing large numbers to smaller ones. Theory: if

Lowest Common Multiple (LCM) Definition: The LCM of two whole numbers a and b

Lowest Common Multiple (LCM) Methods for finding LCM(a, b) (II) Factorization method – a

Lowest Common Multiple (LCM) Methods for finding LCM(a, b) (II) Factorization method – Another

Lowest Common Multiple (LCM) Methods for finding LCM(a, b) (III) “Mechanical method” – using

Lowest Common Multiple (LCM) Example: Your neighbor has two dogs - one named Sparky

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Section 5. 2 Greatest Common Factors and Lowest Common Multiples

Greatest Common Factor (GCF) Definition: The GCF of two whole numbers a and b is the largest whole number c that can divide into both a and b. Methods for finding GCF(a, b) (I) Set intersection method – depends on listing all factors of a and b. example: to find the GCF(12, 18) factors of 12: 1, 2, 3, 4, 6, 12.

Greatest Common Factor (GCF) Methods for finding GCF(a, b) (method I is straight forward and easy to learn but slow for larger numbers) (II) Factorization method – depends on the complete factorization of a and b. example: to find the GCF(24, 180) 24 = 2× 2× 2× 3, 180 = 2× 2× 3× 3× 5 and by the “Greedy algorithm”, we pick all the prime factors common to both numbers (including multiplicities) GCF(24, 180) = 2× 2× 3

One more exercise Find the GCF(882, 924) (click for answer. ) Answer: 882 = 2× 3× 3× 7× 7 924 = 2× 2× 3× 7× 11 Hence GCF(882, 924) = 2× 3× 7 = 42

(method II is still difficult to use if the numbers are too large and difficult to factorize. ) (III) Euclidean Algorithm – depends on reducing large numbers to smaller ones. Theory: if a > b, then GCF(a, b) = GCF(a - b, b) in other words, we can reduce the larger number by subtracting it with the smaller one, without changing the GCF. (click to scroll up)

(III) Euclidean Algorithm – depends on reducing large numbers to smaller ones. Theory: if a > b, then GCF(a, b) = GCF(a - b, b) in other words, we can reduce the larger number by subtracting it with the smaller one, without changing the GCF. Example: GCF(481, 296) = GCF(481 – 296, 296) = GCF(185, 296 – 185) = GCF(185, 111) = GCF(74, 37) = GCF(37, 37) = 37

Lowest Common Multiple (LCM) Definition: The LCM of two whole numbers a and b is the smallest whole number c that is divisible by both a and b. Methods for finding LCM(a, b) (I) Set intersection method – depends on listing small multiples of a and b. example: to find the LCM(15, 24) multiples of 15: multiples of 24:

Lowest Common Multiple (LCM) Methods for finding LCM(a, b) (II) Factorization method – a “thrifty” algorithm example: to find the LCM(875, 1225) 875 = 5× 5× 5× 7 1225 = 5× 5× 7× 7 Analogy: Your math teacher requires you to bring 3 pencils and 1 highlighter to class, your History teacher requires you to bring 2 pencils and 2 highlighters. What is the minimum set of pens you need to bring if you have both classes in one day? Ans: 3 pencils and 2 highlighters.

Lowest Common Multiple (LCM) Methods for finding LCM(a, b) (II) Factorization method – Another example: to find the LCM(60, 126) 60 = 2× 2× 3× 5 126 = 2× 3× 3× 7 Answer: 2× 2× 3× 3× 5× 7

Lowest Common Multiple (LCM) Methods for finding LCM(a, b) (III) “Mechanical method” – using a formula a´b LCM (a, b) = GCF (a, b) Example: Find the LCM of 481 and 296. Answer: we have calculated before that GCF(481, 296) = 37, therefore LCM(481, 296) = (481× 296) ÷ 37 = 3848

Lowest Common Multiple (LCM) Example: Your neighbor has two dogs - one named Sparky and one named Skippy. They both like to bark and both have a pattern. Sparky will bark for a while every 15 minutes once started, and Skippy does the same every 18 minutes. One night they both started barking at 9 pm because a cat passed by. When did they bark together again that night?