m81272 Find Multiples and Factors By the end of this section, you will be able to: Identify multiples of numbers Use common divisibility tests Find all the factors of a number Identify prime and composite numbers f8af279f-9990-4610-b634-0ba844760dea Before you get started, take this readiness quiz. Which of the following numbers are counting numbers (natural numbers)? 0,4,215If you missed this problem, review . 4 and 215 Find the sum of 3,5, and 7.If you missed the problem, review . 15

Annie is counting the shoes in her closet. The shoes are matched in pairs, so she doesn’t have to count each one. She counts by twos: 2,4,6,8,10,12. She has 12 shoes in her closet. The numbers 2,4,6,8,10,12 are called multiples of 2. Multiples of 2 can be written as the product of a counting number and 2. The first six multiples of 2 are given below. 1⋅2=22⋅2=43⋅2=64⋅2=85⋅2=106⋅2=12 A multiple of a number is the product of the number and a counting number. So a multiple of 3 would be the product of a counting number and 3. Below are the first six multiples of 3.1⋅3=32⋅3=63⋅3=94⋅3=125⋅3=156⋅3=18 We can find the multiples of any number by continuing this process. shows the multiples of 2 through 9 for the first twelve counting numbers. Counting Number 1 2 3 4 5 6 7 8 9 10 11 12 Multiples of2 2 4 6 8 10 12 14 16 18 20 22 24 Multiples of3 3 6 9 12 15 18 21 24 27 30 33 36 Multiples of4 4 8 12 16 20 24 28 32 36 40 44 48 Multiples of5 5 10 15 20 25 30 35 40 45 50 55 60 Multiples of6 6 12 18 24 30 36 42 48 54 60 66 72 Multiples of7 7 14 21 28 35 42 49 56 63 70 77 84 Multiples of8 8 16 24 32 40 48 56 64 72 80 88 96 Multiples of9 9 18 27 36 45 54 63 72 81 90 99 108 A number is a multiple of n if it is the product of a counting number and n. Recognizing the patterns for multiples of 2,5,10,and3 will be helpful to you as you continue in this course. Doing the Manipulative Mathematics activity “Multiples” will help you develop a better understanding of multiples. shows the counting numbers from 1 to 50. Multiples of 2 are highlighted. Do you notice a pattern?

Multiples of 2 between 1 and 50

The last digit of each highlighted number in is either 0,2,4,6,or8. This is true for the product of 2 and any counting number. So, to tell if any number is a multiple of 2 look at the last digit. If it is 0,2,4,6,or8, then the number is a multiple of 2. Determine whether each of the following is a multiple of 2: ⓐ 489 ⓑ 3,714 ⓐ Is 489 a multiple of 2? Is the last digit 0, 2, 4, 6, or 8? No. 489 is not a multiple of 2. ⓑ Is 3,714 a multiple of 2? Is the last digit 0, 2, 4, 6, or 8? Yes. 3,714 is a multiple of 2. Determine whether each number is a multiple of 2: ⓐ 678 ⓑ 21,493 ⓐ yes ⓑ no Determine whether each number is a multiple of 2: ⓐ 979 ⓑ 17,780 ⓐ no ⓑ yes Now let’s look at multiples of 5. highlights all of the multiples of 5 between 1 and 50. What do you notice about the multiples of 5?

Multiples of 5 between 1 and 50

All multiples of 5 end with either 5 or 0. Just like we identify multiples of 2 by looking at the last digit, we can identify multiples of 5 by looking at the last digit. Determine whether each of the following is a multiple of 5: ⓐ 579 ⓑ 880 ⓐ Is 579 a multiple of 5? Is the last digit 5 or 0? No. 579 is not a multiple of 5. ⓑ Is 880 a multiple of 5? Is the last digit 5 or 0? Yes. 880 is a multiple of 5. Determine whether each number is a multiple of 5. ⓐ 675 ⓑ 1,578 ⓐ yes ⓑ no Determine whether each number is a multiple of 5. ⓐ 421 ⓑ 2,690 ⓐ no ⓑ yes highlights the multiples of 10 between 1 and 50. All multiples of 10 all end with a zero.

Multiples of 10 between 1 and 50

Determine whether each of the following is a multiple of 10: ⓐ 425 ⓑ 350 ⓐ Is 425 a multiple of 10? Is the last digit zero? No. 425 is not a multiple of 10. ⓑ Is 350 a multiple of 10? Is the last digit zero? Yes. 350 is a multiple of 10. Determine whether each number is a multiple of 10: ⓐ 179 ⓑ 3,540 ⓐ no ⓑ yes Determine whether each number is a multiple of 10: ⓐ 110 ⓑ 7,595 ⓐ yes ⓑ no highlights multiples of 3. The pattern for multiples of 3 is not as obvious as the patterns for multiples of 2,5,and10.

Multiples of 3 between 1 and 50

Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit. The pattern for multiples of 3 is based on the sum of the digits. If the sum of the digits of a number is a multiple of 3, then the number itself is a multiple of 3. See . Multiple of 3 3 6 9 12 15 18 21 24 Sum of digits 3 6 9 1+23 1+56 1+89 2+13 2+46 Consider the number 42. The digits are 4 and 2, and their sum is 4+2=6. Since 6 is a multiple of 3, we know that 42 is also a multiple of 3. Determine whether each of the given numbers is a multiple of 3: ⓐ 645 ⓑ 10,519 ⓐ Is 645 a multiple of 3? Find the sum of the digits. 6+4+5=15 Is 15 a multiple of 3? Yes. If we're not sure, we could add its digits to find out. We can check it by dividing 645 by 3. 645÷3 The quotient is 215. 3⋅215=645 ⓑ Is 10,519 a multiple of 3? Find the sum of the digits. 1+0+5+1+9=16 Is 16 a multiple of 3? No. So 10,519 is not a multiple of 3 either.. 645÷3 We can check this by dividing by 10,519 by 3. 3,506R1310,519 When we divide 10,519 by 3, we do not get a counting number, so 10,519 is not the product of a counting number and 3. It is not a multiple of 3. Determine whether each number is a multiple of 3: ⓐ 954 ⓑ 3,742 ⓐ yes ⓑ no Determine whether each number is a multiple of 3: ⓐ 643 ⓑ 8,379 ⓐ no ⓑ yes Look back at the charts where you highlighted the multiples of 2, of 5, and of 10. Notice that the multiples of 10 are the numbers that are multiples of both 2 and 5. That is because 10=2⋅5. Likewise, since 6=2⋅3, the multiples of 6 are the numbers that are multiples of both 2 and 3.

Another way to say that 375 is a multiple of 5 is to say that 375 is divisible by 5. In fact, 375÷5 is 75, so 375 is 5⋅75. Notice in that 10,519 is not a multiple 3. When we divided 10,519 by 3 we did not get a counting number, so 10,519 is not divisible by 3. If a number m is a multiple of n, then we say that m is divisible by n. Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests. summarizes divisibility tests for some of the counting numbers between one and ten. Divisibility Tests A number is divisible by 2 if the last digit is 0,2,4,6,or8 3 if the sum of the digits is divisible by 3 5 if the last digit is 5 or 0 6 if divisible by both 2 and 3 10 if the last digit is 0 Determine whether 1,290 is divisible by 2,3,5,and10. applies the divisibility tests to 1,290. In the far right column, we check the results of the divisibility tests by seeing if the quotient is a whole number. Divisible by…? Test Divisible? Check 2 Is last digit 0,2,4,6,or8? Yes. yes 1290÷2=645 3 Is sum of digits divisible by3?1+2+9+0=12 Yes. yes 1290÷3=430 5 Is last digit 5 or 0? Yes. yes 1290÷5=258 10 Is last digit 0? Yes. yes 1290÷10=129 Thus, 1,290 is divisible by 2,3,5,and10. Determine whether the given number is divisible by 2,3,5,and10. 6240 Divisible by 2, 3, 5, and 10 Determine whether the given number is divisible by 2,3,5,and10. 7248 Divisible by 2 and 3, not 5 or 10. Determine whether 5,625 is divisible by 2,3,5,and10. applies the divisibility tests to 5,625 and tests the results by finding the quotients. Divisible by…? Test Divisible? Check 2 Is last digit 0,2,4,6,or8? No. no 5625÷2=2812.5 3 Is sum of digits divisible by3?5+6+2+5=18 Yes. yes 5625÷3=1875 5 Is last digit is 5 or 0? Yes. yes 5625÷5=1125 10 Is last digit 0? No. no 5625÷10=562.5 Thus, 5,625 is divisible by 3 and 5, but not 2, or 10. Determine whether the given number is divisible by2,3,5,and10. 4962 Divisible by 2 and 3, not 5 or 10. Determine whether the given number is divisible by2,3,5,and10. 3765 Divisible by 3 and 5.

There are often several ways to talk about the same idea. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. We know that 72 is the product of 8 and 9, so we can say 72 is a multiple of 8 and 72 is a multiple of 9. We can also say 72 is divisible by 8 and by 9. Another way to talk about this is to say that 8 and 9 are factors of 72. When we write 72=8⋅9 we can say that we have factored 72. In the expression a⋅b, both a and b are called factors. If a⋅b=m, and both a and b are integers, then aandb are factors of m, and m is the product of aandb. In algebra, it can be useful to determine all of the factors of a number. This is called factoring a number, and it can help us solve many kinds of problems.Doing the Manipulative Mathematics activity “Model Multiplication and Factoring” will help you develop a better understanding of multiplication and factoring.For example, suppose a choreographer is planning a dance for a ballet recital. There are 24 dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage. In how many ways can the dancers be put into groups of equal size? Answering this question is the same as identifying the factors of 24. summarizes the different ways that the choreographer can arrange the dancers. Number of Groups Dancers per Group Total Dancers 1 24 1⋅24=24 2 12 2⋅12=24 3 8 3⋅8=24 4 6 4⋅6=24 6 4 6⋅4=24 8 3 8⋅3=24 12 2 12⋅2=24 24 1 24⋅1=24 What patterns do you see in ? Did you notice that the number of groups times the number of dancers per group is always 24? This makes sense, since there are always 24 dancers. You may notice another pattern if you look carefully at the first two columns. These two columns contain the exact same set of numbers—but in reverse order. They are mirrors of one another, and in fact, both columns list all of the factors of 24, which are: 1,2,3,4,6,8,12,24 We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with 1. If the quotient is also a counting number, then the divisor and the quotient are factors of the number. We can stop when the quotient becomes smaller than the divisor. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor. If the quotient is a counting number, the divisor and quotient are a pair of factors. If the quotient is not a counting number, the divisor is not a factor. List all the factor pairs. Write all the factors in order from smallest to largest. Find all the factors of 72. Divide 72 by each of the counting numbers starting with 1. If the quotient is a whole number, the divisor and quotient are a pair of factors. The next line would have a divisor of 9 and a quotient of 8. The quotient would be smaller than the divisor, so we stop. If we continued, we would end up only listing the same factors again in reverse order. Listing all the factors from smallest to greatest, we have 1,2,3,4,6,8,9,12,18,24,36,and72 Find all the factors of the given number: 96 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 Find all the factors of the given number: 80 1, 2, 4, 5, 8, 10, 16, 20, 40, 80

Some numbers, like 72, have many factors. Other numbers, such as 7, have only two factors: 1 and the number. A number with only two factors is called a prime number. A number with more than two factors is called a composite number. The number 1 is neither prime nor composite. It has only one factor, itself. A prime number is a counting number greater than 1 whose only factors are 1 and itself. A composite number is a counting number that is not prime. lists the counting numbers from 2 through 20 along with their factors. The highlighted numbers are prime, since each has only two factors.

Factors of the counting numbers from 2 through 20, with prime numbers highlighted

The prime numbers less than 20 are 2,3,5,7,11,13,17,and19. There are many larger prime numbers too. In order to determine whether a number is prime or composite, we need to see if the number has any factors other than 1 and itself. To do this, we can test each of the smaller prime numbers in order to see if it is a factor of the number. If none of the prime numbers are factors, then that number is also prime. Test each of the primes, in order, to see if it is a factor of the number. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime. Identify each number as prime or composite: ⓐ 83 ⓑ 77 ⓐ Test each prime, in order, to see if it is a factor of 83, starting with 2, as shown. We will stop when the quotient is smaller than the divisor. Prime Test Factor of 83? 2 Last digit of 83 is not 0,2,4,6,or8. No. 3 8+3=11, and 11 is not divisible by 3. No. 5 The last digit of 83 is not 5 or 0. No. 7 83÷7=11.857…. No. 11 83÷11=7.545… No. We can stop when we get to 11 because the quotient (7.545…) is less than the divisor. We did not find any prime numbers that are factors of 83, so we know 83 is prime. ⓑ Test each prime, in order, to see if it is a factor of 77. Prime Test Factor of 77? 2 Last digit is not 0,2,4,6,or8. No. 3 7+7=14, and 14 is not divisible by 3. No. 5 the last digit is not 5 or 0. No. 7 77÷7=11 Yes. Since 77 is divisible by 7, we know it is not a prime number. It is composite. Identify the number as prime or composite: 91 composite Identify the number as prime or composite: 137 prime The Links to Literacy activities One Hundred Hungry Ants, Spunky Monkeys on Parade and A Remainder of One will provide you with another view of the topics covered in this section. Divisibility Rules Factors Ex 1: Determine Factors of a Number Ex 2: Determine Factors of a Number Ex 3: Determine Factors of a Number

Divisibility Tests A number is divisible by 2 if the last digit is 0, 2, 4, 6, or 8 3 if the sum of the digits is divisible by 3 4 if the last two digits are a number divisible by 4 5 if the last digit is 5 or 0 6 if divisible by both 2 and 3 10 if the last digit is 0 Factors If a⋅b=m, then a and b are factors of m, and m is the product of a and b. Find all the factors of a counting number. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor. If the quotient is a counting number, the divisor and quotient are a pair of factors. If the quotient is not a counting number, the divisor is not a factor. List all the factor pairs. Write all the factors in order from smallest to largest. Determine if a number is prime. Test each of the primes, in order, to see if it is a factor of the number. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.

Identify Multiples of Numbers In the following exercises, list all the multiples less than 50 for the given number. 2 2, 4, 6, 8, 10 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48 3 4 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 5 6 6, 12, 18, 24, 30, 36, 42, 48 7 8 8, 16, 24, 32, 40, 48 9 10 10, 20, 30, 40 12 Use Common Divisibility Tests In the following exercises, use the divisibility tests to determine whether each number is divisible by 2,3,4,5,6,and10. 84 Divisible by 2, 3, 4, 6 96 75 Divisible by 3, 5 78 168 Divisible by 2, 3, 4, 6 264 900 Divisible by 2, 3, 4, 5, 6, 10 800 896 Divisible by 2, 4 942 375 Divisible by 3, 5 750 350 Divisible by 2, 5, 10 550 1430 Divisible by 2, 5, 10 1080 22,335 Divisible by 3, 5 39,075 Find All the Factors of a Number In the following exercises, find all the factors of the given number. 36 1, 2, 3, 4, 6, 9, 12, 18, 36 42 60 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 48 144 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72,144 200 588 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588 576 Identify Prime and Composite Numbers In the following exercises, determine if the given number is prime or composite. 43 prime 67 39 composite 53 71 prime 119 481 composite 221 209 composite 359 667 composite 1771

Banking Frank’s grandmother gave him $100 at his high school graduation. Instead of spending it, Frank opened a bank account. Every week, he added $15 to the account. The table shows how much money Frank had put in the account by the end of each week. Complete the table by filling in the blanks. Weeks after graduation Total number of dollars Frank put in the account Simplified Total 0 100 100 1 100+15 115 2 100+15⋅2 130 3 100+15⋅3 4 100+15⋅[] 5 100+[] 6 20 x Banking In March, Gina opened a Christmas club savings account at her bank. She deposited $75 to open the account. Every week, she added $20 to the account. The table shows how much money Gina had put in the account by the end of each week. Complete the table by filling in the blanks. Weeks after opening the account Total number of dollars Gina put in the account Simplified Total 0 75 75 1 75+20 95 2 75+20⋅2 115 3 75+20⋅3 4 75+20⋅[] 5 75+[] 6 20 x

If a number is divisible by 2 and by 3, why is it also divisible by 6? What is the difference between prime numbers and composite numbers?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

multiple of a number A number is a multiple of n if it is the product of a counting number and n. divisibility If a number m is a multiple of n, then we say that m is divisible by n. prime number A prime number is a counting number greater than 1 whose only factors are 1 and itself. composite number A composite number is a counting number that is not prime.