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2.4 Find multiples and factors  (Page 4/9)

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Find all the factors of 72 .

Solution

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 figure shows a table with ten rows and four columns. The first row is a header row and labels the rows “Dividend”, “Divisor”, “Quotient”, and “Factors”. Under the “Dividend” column all rows show the number 72. In the second row the “Divisor” column is 1, the “Quotient” column is 72 and the “Factors” column is 1 and 72. In the third row the “Divisor” column is 2, the “Quotient” column is 36 and the “Factors” column is 2 and 36. In the fourth row the “Divisor” column is 3, the “Quotient” column is 24 and the “Factors” column is 3 and 24. In the fifth row the “Divisor” column is 4, the “Quotient” column is 18 and the “Factors” column is 4 and 18. In the sixth row the “Divisor” column is 5, the “Quotient” column is 14.4 and the “Factors” column is blank. In the seventh row the “Divisor” column is 6, the “Quotient” column is 12 and the “Factors” column is 6 and 12. In the eighth row the “Divisor” column is 7, the “Quotient” column is about 10.29 and the “Factors” column is blank. In the ninth row the “Divisor” column is 8, the “Quotient” column is 9 and the “Factors” column is 8 and 9. In the tenth row the “Divisor” column is 9, the “Quotient” column is 8 and the “Factors” column is 9 and 8.

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 , and 72

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Find all the factors of the given number:

96

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

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Find all the factors of the given number:

80

1, 2, 4, 5, 8, 10, 16, 20, 40, 80

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Identify prime and composite numbers

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.

Prime numbers and composite numbers

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.

[link] lists the counting numbers from 2 through 20 along with their factors. The highlighted numbers are prime, since each has only two factors.

This figure shows a table with twenty rows and three columns. The first row is a header row. It labels the columns as “Number”, “Factor” and “Prime or composite?” The second row lists the number 2, in red, under the “Number” column, the numbers 1 and 2 under the “Factors” column and the word prime under the “Prime or Composite?” column. The third row lists the number 3, in red, under the “Number” column, the numbers 1 and 3 under the “Factors” column and the word prime under the “Prime or Composite?” column. The fourth row lists the number 4 under the “Number” column, the numbers 1, 2 and 4 under the “Factors” column and the word composite under the “Prime or Composite?” column. The fifth row lists the number 5, in red, under the “Number” column, the numbers 1 and 5 under the “Factors” column and the word prime under the “Prime or Composite?” column. The sixth row lists the number 6 under the “Number” column, the numbers 1, 2, 3 and 6 under the “Factors” column and the word composite under the “Prime or Composite?” column. The seventh row lists the number 7, in red, under the “Number” column, the numbers 1 and 7 under the “Factors” column and the word prime under the “Prime or Composite?” column. The eighth row lists the number 8 under the “Number” column, the numbers 1, 2, 4 and 8 under the “Factors” column and the word composite under the “Prime or Composite?” column. The ninth row lists the number 9 under the “Number” column, the numbers 1, 3 and 9 under the “Factors” column and the word composite under the “Prime or Composite?” column. The tenth row lists the number 10 under the “Number” column, the numbers 1, 2, 5 and 10 under the “Factors” column and the word composite under the “Prime or Composite?” column. The eleventh row lists the number 11, in red, under the “Number” column, the numbers 1 and 11 under the “Factors” column and the word prime under the “Prime or Composite?” column. The twelfth row lists the number 12 under the “Number” column, the numbers 1, 2, 3, 4, 6 and 12 under the “Factors” column and the word composite under the “Prime or Composite?” column. The thirteenth row lists the number 13, in red, under the “Number” column, the numbers 1 and 13 under the “Factors” column and the word prime under the “Prime or Composite?” column. The fourteenth row lists the number 14 under the “Number” column, the numbers 1, 2, 7 and 14 under the “Factors” column and the word composite under the “Prime or Composite?” column. The fifteenth row lists the number 15 under the “Number” column, the numbers 1, 2, 3, 5 and 15 under the “Factors” column and the word composite under the “Prime or Composite?” column. The sixteenth row lists the number 16 under the “Number” column, the numbers 1, 2, 4, 8 and 16 under the “Factors” column and the word composite under the “Prime or Composite?” column. The seventeenth row lists the number 17, in red, under the “Number” column, the numbers 1 and 17 under the “Factors” column and the word prime under the “Prime or Composite?” column. The eighteenth row lists the number 18 under the “Number” column, the numbers 1, 2, 3, 6, 9 and 18 under the “Factors” column and the word composite under the “Prime or Composite?” column. The nineteenth row lists the number 19, in red, under the “Number” column, the numbers 1 and 19 under the “Factors” column and the word prime under the “Prime or Composite?” column. The twentieth row lists the number 20 under the “Number” column, the numbers 1, 2, 4, 5, 10 and 20 under the “Factors” column and the word composite under the “Prime or Composite?” column.
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 , and 19 . 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.

Determine if a number is prime.

  1. Test each of the primes, in order, to see if it is a factor of the number.
  2. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found.
  3. 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:

  1. ⓐ 83
  2. ⓑ 77

Solution

ⓐ Test each prime, in order, to see if it is a factor of 83 by each of the prime numbers, 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 , or 8 . 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 , or 8 . 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 ÷ 11 = 7 Yes.

Since 77 is divisible by 7 , we know it is not a prime number. It is composite.

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Identify the number as prime or composite:

91

composite

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Identify the number as prime or composite:

137

prime

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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.
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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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