Prime Numbers and Prime Factorization next Taking the Fear out of Math #5 Prime Numbers and Prime Factorization 3 × 1 3 © Math As A Second Language All Rights Reserved

Introduction to Prime Numbers next next 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... © Math As A Second Language All Rights Reserved

next next 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. © Math As A Second Language All Rights Reserved

Using the properties of whole numbers… next next 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 © Math As A Second Language All Rights Reserved

neither is their product. next next 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, 45. . . 7, 14, 21, 28, 35, 42, 49, 56, 63. . . As before, neither 4 or 9 is a multiple of either 5 or 7. However, this time neither is their product. © Math As A Second Language All Rights Reserved

Therefore, the only way a number next next next 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. © Math As A Second Language All Rights Reserved

Definitions next next next 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. © Math As A Second Language All Rights Reserved

Composite numbers can be factored in several different ways. next next 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 © Math As A Second Language All Rights Reserved

next next 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 © Math As A Second Language All Rights Reserved

On the other hand, starting with next next next 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. © Math As A Second Language All Rights Reserved

next next 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. © Math As A Second Language All Rights Reserved

Prime Factorization and Least Common Multiples next next next 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. © Math As A Second Language All Rights Reserved

next next next 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. © Math As A Second Language All Rights Reserved

Therefore, the least common multiple of next next next next 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). © Math As A Second Language All Rights Reserved

missing, it won’t be a multiple of 8. next 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. © Math As A Second Language All Rights Reserved

determined that 40 is the least common multiple of 8 and 10. next next 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… © Math As A Second Language All Rights Reserved

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. next next next 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. © Math As A Second Language All Rights Reserved

However, as we will now show, it isn’t the least common multiple. next next 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. © Math As A Second Language All Rights Reserved

To find the least common multiple, for 2, 3, 4, 5, 6, 7, 8, 9, and 10 next next 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”). © Math As A Second Language All Rights Reserved

And since 2 and 3 are both prime And since 1 is the smallest non zero next next next 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 ). © Math As A Second Language All Rights Reserved

4 × ___, or in terms of prime factorization, 2 × 2 × ___ . next next next 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). © Math As A Second Language All Rights Reserved

The problem now is that 2 × 3 × 2 does not next next next 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) . © Math As A Second Language All Rights Reserved

6 × ___, or in terms of prime factorization, 2 × 3 × ___. next next next 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. © Math As A Second Language All Rights Reserved

The problem now is that while 7 is a next next next 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) . © Math As A Second Language All Rights Reserved

8 × ___, or in terms of prime factorization, 2 × 2 × 2 × ___. next next next 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). © Math As A Second Language All Rights Reserved

9 × ___, or in terms of prime factorization, 3 × 3 × ___. next next next 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). © Math As A Second Language All Rights Reserved

10 × ___, or in terms of prime factorization, 2 × 5 × ___. next next 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. © Math As A Second Language All Rights Reserved

next next 2 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 © Math As A Second Language All Rights Reserved

next 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. © Math As A Second Language All Rights Reserved

For example, if we write 2,520 in the form 2 × 2 × 2 × 3 × 3 × 5 × 7 next next 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. © Math As A Second Language All Rights Reserved

Notice that we can rearrange the factors next next 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. © Math As A Second Language All Rights Reserved

In our next lessons, we will show how prime factors and least common multiples are related to the arithmetic of fractions. © Math As A Second Language All Rights Reserved