1. Prime Numbers and Prime Factorization
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Taking the Fear
out of Math #5
Prime Numbers and Prime Factorization
3 × 1 3
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2. Introduction to Prime Numbers
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Introduction to Prime Numbers
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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6, 12, 18, 24, 30, 36, 42, 48...
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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4. Using the properties of whole numbers…
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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…
4 × 9
= (2 × 2) × (3 × 3)
= 2 × 2 × 3 × 3
= 2 × 3 × 2 × 3
= (2 × 3) × (2 × 3)
= 6 × 6
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5. neither is their product.
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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).
5, 10, 15, 20, 25, 30, 35, 40,
7, 14, 21, 28, 35, 42, 49, 56,
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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6. Therefore, the only way a number
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What happened here is that, unlike 6, neither 5 nor 7 had factors other than 1 and itself.1
Therefore, the only way a number
can 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.
note
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.
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7. Definitions
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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.
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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8. Composite numbers can be factored in several different ways.
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An Introduction to Prime Factorization
Composite numbers can be factored in several different ways.
For example, we may factor 12 as…
1 × 12
and
2 × 2 × 3
2 × 6
3 × 4
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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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10. On the other hand, starting with
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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
12 = 2 × 6
= 3 × (2 × 2)
= 2 × (2 × 3)
= 3 × 2 × 2
= 2 × 2 × 3
Notice while the factors appear in a different order, the prime factorization
is the same.
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Key Point
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.
note
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 as
12 = 2 × 2 × 3 = 2 × 2 × 3 × 1 = 2 × 2 × 3 × 1 × 1, etc.
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12. Prime Factorization and Least Common Multiples
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Prime Factorization and
Least 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.
Thus, any multiple of 8 must have the form
2 × 2 × 2 × N where N is any whole number.3
note
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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14. Therefore, the least common multiple of
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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).
8 = 2 × 2 × 2
2 × 2 × 2
10 = 2 × 5
2 × 5
= 40
Therefore, the least common multiple of
8 and 10 is 2 × 2 × 2 × 5 (= 40).
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15. missing, it won’t be a multiple of 8.
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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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16. determined that 40 is the least common multiple of 8 and 10.
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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 10
and 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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17. As a case in point, let’s use prime
However, as we will discuss in more detail in our presentation on fractions, this is not always the case.
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As a case in point, let’s use prime
factorization 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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18. However, as we will now show, it isn’t the least common multiple.
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Note
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.
However, as we will now show, it isn’t
the least common multiple.
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19. To find the least common multiple, for 2, 3, 4, 5, 6, 7, 8, 9, and 10
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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 דany whole number”).
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20. And since 2 and 3 are both prime And since 1 is the smallest non zero
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2 × ___
► To be a multiple of 3, the number we are looking for must have the form
3 × ___ .
And since 2 and 3 are both prime
numbers, 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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21. 4 × ___, or in terms of prime factorization, 2 × 2 × ___ .
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2 × 3 × ___
► To be a multiple of 4, the number we are looking for must have the form
4 × ___, or in terms of prime factorization,
2 × 2 × ___ .
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 common
multiple of 2, 3, and 4 is 2 × 2 × 3 × 1 (or 12).
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22. The problem now is that 2 × 3 × 2 does not
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2 × 3 × 2 × ___
► To be a multiple of 5, the number we are looking for must have the form
5 × ___.
The problem now is that 2 × 3 × 2 does not
contain 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) .
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23. 6 × ___, or in terms of prime factorization, 2 × 3 × ___.
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2 × 3 × 2 × 5 × ___
► To be a multiple of 6, the number we are looking for must have the form
6 × ___, or in terms of prime factorization,
2 × 3 × ___.
However, since 60 already contains 2 × 3
as 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.
note
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.
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24. The problem now is that while 7 is a
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2 × 3 × 2 × 5 × ___
► To be a multiple of 7, the number we are looking for must have the form
7 × ___.
The problem now is that while 7 is a
prime 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) .
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25. 8 × ___, or in terms of prime factorization, 2 × 2 × 2 × ___.
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2 × 3 × 2 × 5 × 7 × ___
► To be a multiple of 8, the number we are looking for must have the form
8 × ___, or in terms of prime factorization, 2 × 2 × 2 × ___.
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).
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26. 9 × ___, or in terms of prime factorization, 3 × 3 × ___.
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2 × 3 × 2 × 5 × 7 × 2 ___
► To be a multiple of 9, the number we are looking for must have the form
9 × ___, or in terms of prime factorization, 3 × 3 × ___.
However, since 840 contains 3 but not
3 × 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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27. 10 × ___, or in terms of prime factorization, 2 × 5 × ___.
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2 × 3 × 2 × 5 × 7 × 2 × 3 ___
► To be a multiple of 10, the number we are looking for must have the form
10 × ___, or in terms of prime factorization, 2 × 5 × ___.
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.
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2 × 3 × 2 × 5 × 7 × 2 × 3
× 3
× 2
× 5
× 7
× 2
× 3
= 2,520
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.
= 2,520
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Note
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.
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30. For example, if we write 2,520 in the form 2 × 2 × 2 × 3 × 3 × 5 × 7
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Note
By writing 2,520 as a product of prime numbers we can see immediately whether a given number is a divisor of 2,520.
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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31. Notice that we can rearrange the factors
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Note
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).
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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32. In our next lessons, we will show how prime factors and
least common multiples are related to the arithmetic of fractions.
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