8.1 Factors and Multiples

the multiples of 21 are , we cannot list the full set of multiples of 21 because

21, 42, 63, 84, 105, 126 the list is infinitely long

factor 18 as we can factor 18 even further...

9 x 2: 18 = 9 x 2, because we can factor 9 as 3 x 3, so 18 = 3 x 3 x 2

to summarize, if A, B, and C are counting numbers and if A = B X c then

A is a multiple of B A is multiple of C B Is a factor of A c is a factor of A

if A, B , and C are counting numbers such that A = B x C then we say we can also say

A is a multiple of B and of C and that B and C are factors or divisors of A B and C divide A and A is divisible by B and C sometimes we say that A is evenly divisible by B and C so the factors of 21 are 1, 3, 7, 21

we often use factors and multiples when we

work with equivalent fractions.

number theory

study of whole numbers (no fractions, decimals)

so far we've used the word factor as a noun, we can also the word as a what does it mean to factor A

verb. if A is a counting number, then to factor A means to write A as a product of two or more counting numbers each of which is less than A

how are the concepts of factors and multiples closely linked....

a counting number A is a multiple of a counting number B exactly when B is a factor of A.

find all the factors of 40 using the counting numbers 1-8 you don't have to check if 9 , 10, 11, and 12, and so on divide by 40 ..why? the full list of factors of 40 is , what do you notice about every counting number?

divide 40 by all counting numbers, recording those numbers that divide 40 and recording the corresponding quotients after dividing 40 by 1 through 8 and verifying that 1, 2, 4, 5 and 8 are the only factors of 40 up to 8, because any counting number larger than 8 that divides 40 would have a quotient less than 5 and would already have to appear as one of the factors listed previously is 1 2 4 5 8 10 20 40 , every counting number except ` must have at least two distinct factors - namely 1 and itself

what is one way to find all the factors of a counting number...

divide the number by all the counting numbers smaller than it to see which ones divide the number evenly, without a remainder keep track of the quotients because the whole number quotients will also be factors

what are factors and multiples and why do we have these concepts?

factors and multiples arise when we go beyond solving an individual multiplication or division problem and instead consider the different ways that numbers can be built up or broken down through multiplication and division

we commonly use the concepts of it is useful to include 0 as well; how can we describe 0

factors and multiples in the context of counting numbers describe 0 as divisible by every counting number, or as a multiple of every whole number

it's easy to get the concepts of factors and multiples confused,.. building up vs breaking down

factors of a number are the numbers you get from writing the original number as a product from breaking down the number by dividing it multiples of a number are the numbers you get by multiplying the number by counting numbers, but building up the number by multiplying it

mathematicians have long been intrigued by factors and multiples

for what they reveal about the structure of the system of counting numbers

finding the single number c such that 30= 5 x c finding the single number A A = 2 x 8

is a division problem is a multiplication problem

finding all the counting numbers B and C 30 = B x C finding all the counting numbers A, for some counting number B A = 3 x 8

is the problem of finding all the factors of 30. is the problem of finding all the multiples of 8

for a small number,

it is easy to list all of its factors