8.5 Greatest Common Factor and Least Common Multiple

put 24/26 in simplest form

GCF of 24 and 36 is 12 12 x 2 = 24 3 x 12 -=36, 2/3 is the simplest form of 24 and 26

what do gears, different types of cicadas, and the spirograh drawing toy all have in common?

all have aspects that are related to greatest common factors and least common multiples

to determine the GCF and LCM of counting numbers what two methods could be used

definitions slide method (quicker way)

to put a fraction in simplest form

divide the numerator and denominator of the fraction by the GCF of the numerator and the denominator

in 8.1 we discussed

factors and multiples of individual numbers

how can we use slides flexibly

find the common factor in one step, rather than taking two steps

least common multiiple

if you have two or more counting numbers the LCM of these numbers is the least counting number that is a multiple of all the given numbers

what is the greatest common factor (greatest common divisor)

if you have two or more counting numbers, then the greatest common factor of these numbers is the greatest counting number that is a factor of all the given counting numbers

DEFINITIONS OF GCF AND LCM

one method to finding GCF and LCMS

table 8.2 shows the steps for one way to use the slide method to determine the GCF and LCM of 8800 and 10,000 table 8.3 shows final result of a more efficient way to use the slide method

page 337

add 1/6 and 3/8

pg 338

simplify 24/36

pg 338

to use the slide method

repeatedly find common factors the slide stops when the resulting quotients on the right no longer have any common factor except 1 The GCF is then the product of the factors down the left hand side LCM is the product of the factors down the left hand side of the slide and the numbers in the last row of the slide

what is the GCF of 12 and 18?

the factors of 12 are 1, 2, 3, 4, 6, 12 factors of 18 are 1, 2, 3, 6, 9, 18 common factors of 12 and 18 are the numbers that are common to the two lists 1, 2, 3, 6 the greatest of these numbers is 6 (GCF) of 12 and 18

LCM of 6 and 8

the multiples of 6 are 6 12 18 24 30 36 42 48 54 60 66 72 78 multiples of 8 are 8 16 24 32 40 48 56 64 72 80 LCM of 6 and 8 are the numbers common to the two lists: 24, 48, 72 the least of these numbers is 24 (LCM)

why does the slide method give us the LCM

the product of the numbers on the left in a slide and one of the numbers in the bottom row is equal to the number above it at the top product of the numbers on the left and the numbers in the bottom row of a slide must be a multiple of the initial numbers because we don't repeat the factors that the two initial numbers have in common (factors on the left) we get the smallest possible multiple of the initial numbers

By considering common multiples and common factors of two or more counting numbers,

we arrive at the concepts of greatest common factor and least common multiple,

If we have two or more counting numbers in mind,

we can consider the factors that the two numbers have in common and the multiples that the two numbers have in common

is is not necessary to determine the GCF of the numerator and denominator of a fraction in order to put the fraction in simplest form.

we can keep simplifying the fraction until it can no longer be simplified

by listing successive quotients on the right hand side of the slide,

we ensure we don't repeat a common factor that has already been accounted for

we can use the LCM of the denominators of two fractions to add the fractions. in order to add fractions,

we must first find a common denominator. any common denominator is fine (but the resulting sum will not be in simplest form. this is true even if we use the LCD when adding two fractions, the resulting sum may not be in simplest form ) find LCD because CD of two fractions must be multiples of both denominators, the LCD is the LCM of two denominators

why does the slide method give us the GCF

we repeatedly take out common factors until there are no more common factors left except 1 when we multiply all the common factors we collected, that product should form the GCF by listing successive quotients on the right hand side of the slide, we ensure we don't repeat a common factor that has already been accounted for

greatest common factors and least common multiples can be useful when

working with fractions