Prime Numbers and Prime Factorization © Math As A Second Language All Rights Reserved next #5...

Prime Numbers and

Prime Factorization

© Math As A Second Language All Rights Reserved

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Taking the Fearout of Math

3 × 1

3

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Introduction to Prime Numbersnext

A convenient way to find the least common multiple of two or more numbers is by

using what is called prime factorization.

To get a grasp of what prime numbers are, let’s begin by looking at the multiples of 6.

6, 12, 18, 24, 30, 36, 42, 48...

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Notice that neither 4 nor 9 are on this list but that their product (4 × 9 = 36) is on the list.

To see why this happened notice that 6 can be factored as 3 × 2, and hence every

multiple of 6 has 2 and 3 as factors. Thus, a number is a multiple of 6 if and only if it has 2 and 3 as factors. 4 has 2 as a factor

(that is, 4 = 2 × 2) but not 3, while 9 has 3 as a factor (that is, 9 = 3 × 3) but not 2.

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6, 12, 18, 24, 30, 36, 42, 48...

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However, when we multiply 4 and 9, a factor of 4 (that is, 2) combines with a factor

of 9 (that is, 3) to form 6 as a factor.

Using the properties of whole numbers…

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4 × 9 = (2 × 2) × (3 × 3)

= 2 × 2 × 3 × 3

= 2 × 3 × 2 × 3

= (2 × 3) × (2 × 3)

= 6 × 6

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However, quite a different thing happens when we look at the multiples of either 5 or 7

(which simply happen to be the whole numbers that 6 is between).

As before, neither 4 or 9 is a multiple of either 5 or 7. However, this time

neither is their product.

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5, 10, 15, 20, 25, 30, 35, 40, 45. . .

7, 14, 21, 28, 35, 42, 49, 56, 63. . .

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What happened here is that, unlike 6, neither 5 nor 7 had factors other than 1

and itself.1

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1 Notice that every whole number greater than 1 has at least 2 factors (divisors); namely, 1 and itself. However, while 6 also has 2 and 3 as

additional factors, 5 and 7 have no additional factors.

Therefore, the only way a numbercan be a multiple of 5 is if it is itself divisible by 5, and the only way a

number can be a multiple of 7 is if it is itself divisible by 7.

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A whole number greater than 1 is called a prime number if its only divisors are 1 and itself.

Examples of prime numbers are 2, 3, 5, 7, 11, and 13.

Definitions

A whole number greater than 1 is called a composite number if it is not a prime number. Examples of

composite numbers are 4, 6, 8, 9, 10,12, 14, and 15.

1 is called a unit. It is neither prime nor composite.

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An Introduction to Prime Factorizationnext

Composite numbers can be factored in several different ways.

For example, we may factor 12 as…

1 × 12

2 × 6

3 × 4

and 2 × 2 × 3

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However, if we insist on prime factorization in which every factor is a

prime number, there is only one way that this can be done (except for the order in

which we write the factors).

For example, starting with 12 = 2 × 6, we may rewrite 6 as 2 × 3 and thus obtain…

12 = 2 × 6

= 2 × 2 × 3

= 2 × (2 × 3)

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On the other hand, starting with 12 = 3 × 4, we may rewrite 4 as 2 × 2 and

thus obtain the prime factorization…

12 = 3 × 4

= 3 × 2 × 2

= 3 × (2 × 2)

Notice while the factors appear in a different order, the prime factorization

is the same.

12 = 2 × 6

= 2 × 2 × 3

= 2 × (2 × 3)

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In summary, the principle of prime factorization tells us that a whole number greater than 1 can be written in one and

only one way as a product of prime numbers (except for the order in which the

factor are written), for example,

12 = 2 × 2 × 3 = 2 × 3 × 2 = 3 × 2 × 22.

Key Point

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2 This is one reason why 1 is not considered to be a prime number. More specifically, if 1 was a prime number the prime factorization property

would not apply because 12 could then be factored as12 = 2 × 2 × 3 = 2 × 2 × 3 × 1 = 2 × 2 × 3 × 1 × 1, etc.

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Prime Factorization andLeast Common Multiples

Let’s return to the problem in our previous presentation of the

hot dogs and hot dog buns. The buns come in packages of 8, and the prime

factorization of 8 is 2 × 2 × 2.

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Thus, any multiple of 8 must have the form 2 × 2 × 2 × N where N is any whole number.3

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3 Do not be confused by letting N stand for any whole number. It simply means, for example, that if we choose N to be 7, 2 × 2 × 2 × 7 is the 7th

multiple of 8. And in a similar way, 2 × 2 × 2 × 13 is the 13th multiple of 8.

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On the other hand, since the hot dogs come in packages of 10, and the prime factorization of 10 is 2 × 5, we

see that any multiple of 10 must have the form 2 × 5 × M where M is any

whole number.

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Thus, we see that to be a common multiple of 8 and 10, it must have at least 3 factors of 2 (because 8 = 2 × 2 × 2) and at least 1 factor of 5 (because 10 = 2 × 5).

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8 = 2 × 2 × 2 10 = 2 × 52 × 2 × 2 2 × 5

= 40

Therefore, the least common multiple of 8 and 10 is 2 × 2 × 2 × 5 (= 40).

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Notice that nothing smaller can be a multiple of both 8 and 10 (because if the 5 isn’t present, it won’t be a multiple of 10, and if even one of the factors of 2 is

missing, it won’t be a multiple of 8.

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Thus, without specifically listing the multiples of both 8 and 10, we have

determined that 40 is the least common multiple of 8 and 10.

In this example, it wouldn’t be too tedious to list the multiples of both 8 and

10and see that 40 was the first number that

appeared on both lists of multiples.8, 16, 24, 32, 40, 48…

10, 20, 30, 40, 50, 60…

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next However, as we will discuss in more detail in our presentation on fractions, this

is not always the case.

As a case in point, let’s use primefactorization to find the least common multiple of 2, 3, 4, 5, 6, 7, 8, 9 and 10.

Note

We can always find a common multiple of any number of whole numbers simply by

multiplying all of the numbers.

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Therefore, 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 = 3,628,800 is a common multiple of 2, 3, 4, 5, 6, 7, 8, 9,

and 10.

Note

However, as we will now show, it isn’t the least common multiple.

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To find the least common multiple, for 2, 3, 4, 5, 6, 7, 8, 9, and 10

we know that…

► To be a multiple of 2, the number we are looking for must have the form

2 × ___ (i.e. 2 ×“any whole number”).

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And since 2 and 3 are both primenumbers, to be a multiple of both 2 and 3, the number we are looking or must have

the form 2 × 3 × ___.

And since 1 is the smallest non zero whole number, the least common multiple

of 2 and 3 is 2 × 3 × 1 (or 6 ).

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3 × ___ .

2 × ___

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The problem now is that 2 × 3 has only one factor of 2, which means that to have the least common multiple of 2, 3, and 4,

we need another factor of 2.

In other words, the least commonmultiple of 2, 3, and 4 is 2 × 2 × 3 × 1 (or 12).


4 × ___, or in terms of prime factorization, 2 × 2 × ___ .

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2 × 3 × ___

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The problem now is that 2 × 3 × 2 does notcontain 5 as a prime factor. Therefore, to

convert 2 × 2 × 3 into a common multiple of 2, 3, 4, and 5, we need to multiply

2 × 3 × 2 by 5.

In other words, the least common multiple of 2, 3, 4, and 5 is 2 × 3 × 2 × 5 (or 60) .


5 × ___.

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2 × 3 × 2 × ___

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However, since 60 already contains 2 × 3as a factor, it means that the least common

multiple of 2, 3, 4, 5, and 6 is still 2 × 2 × 3 × 5 (or 60)4.


6 × ___, or in terms of prime factorization, 2 × 3 × ___.

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4 This might be a good reason for why the Babylonians liked to work with 60 as the base of their number system. Namely it is the smallest positive

whole number that is divisible by 2, 3, 4, 5, and 6.

2 × 3 × 2 × 5 × ___

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The problem now is that while 7 is aprime number, but 60 does not contain 7 as

a prime factor. Therefore, to convert 60 into a common multiple of 2, 3, 4, 5, 6, and

7, we need to multiply 2 × 3 × 2 × 5 by 7.

In other words, the least common multiple of 2, 3, 4, 5, and 7 is 2 × 3 × 2 × 5 × 7

(or 420) .


7 × ___.

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2 × 3 × 2 × 5 × ___

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However, since 420 contains 2 × 2 but not 2 × 2 × 2 as a factor, it means that the least

common multiple of 2, 3, 4, 5, 6, 7, and 8 must contain an additional factor of 2.

In other words, the least common multiple of 2, 3, 4, 5, 6, 7, and 8 is

2 × 3 × 2 × 5 × 7 × 2 (or 840).


8 × ___, or in terms of prime factorization, 2 × 2 × 2 × ___.

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2 × 3 × 2 × 5 × 7 × ___

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However, since 840 contains 3 but not3 × 3 as a factor, it means that the least

common multiple of 2, 3, 4, 5, 6, 7, 8 and 9 must contain an additional factor of 3.

In other words, the least common multiple of 2, 3, 4, 5, 6, 7, 8, and 9 is

2 × 3 × 2 × 5 × 7 × 2 × 3 (or 2,520).



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2 × 3 × 2 × 5 × 7 × 2 ___

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Since 2 × 5 is already a factor, the least common multiple of

2, 3, 4, 5, 6, 7, 8, 9, and 10 is also 2,520.



2 × 3 × 2 × 5 × 7 × 2 × 3 ___

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In prime factorization, it is traditional to arrange the factors from least to greatest.

Notice that changing the order of the factors does not change the product.

2 × 3 × 2 × 2× 5 × 7 × 32 × 3 × 2 × 5 × 7 × 2 × 3 = 2,520

= 2,520

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To see how tedious it would be to try to find the least common multiple by listing the multiples of 2, 3, 4, 5, 6, 7, 8, 9 and 10, and then looking for the first number that appeared on each of the lists, notice that 2,520 is the 1,260th multiple of 2, the 840th multiple of 3, the 630th multiple of 4, the 504th multiple of 5, the 420th multiple of 6,

the 360th multiple of 7, the 315th multiple of 8, the 280th multiple of 9, and the 252nd

multiple of 10.

Note

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By writing 2,520 as a product of prime numbers we can see immediately whether

a given number is a divisor of 2,520.

Note

For example, if we write 2,520 in the form 2 × 2 × 2 × 3 × 3 × 5 × 7

We can see that since 35 = 5 × 7, it is a divisor of 2,520.

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Notice that we can rearrange the factors of 2,520 to obtain…

(5 ×7) × (2 × 2 × 2 × 3 × 3) or 35 × 72. In other words 2,520 is the 72nd multiple

of 35 (and the 35th multiple of 72).

Note

On the other hand, 22 is not a divisor of 2,520 because 22 = 2 × 11 and the prime number 11 is not a prime factor of 2,520.

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In our next lessons, we will show how

prime factors and least common multiples

are related to the arithmetic of fractions.


Prime Factors

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