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Study Guides > ALGEBRA / TRIG I

Finding All the Factors of a Number

Learning Outcomes

  • Find all the factors of a number
  • Determine whether a number is prime or composite
There are often several ways to talk about the same idea. So far, we’ve seen that if mm is a multiple of nn, we can say that mm is divisible by nn. We know that 7272 is the product of 88 and 99, so we can say 7272 is a multiple of 88 and 7272 is a multiple of 99. We can also say 7272 is divisible by 88 and by 99. Another way to talk about this is to say that 88 and 99 are factors of 7272. When we write 72=8⋅972=8\cdot 9 we can say that we have factored 7272. The image shows the equation 8 times 9 equals 72. The 8 and 9 are labeled as factors and the 72 is labeled product.

Factors

If a⋅b=ma\cdot b=m, then a and ba\text{ and }b are factors of mm, and mm is the product of a and ba\text{ and }b.
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. For example, suppose a choreographer is planning a dance for a ballet recital. There are 2424 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 2424. The table below summarizes the different ways that the choreographer can arrange the dancers.
Number of Groups Dancers per Group Total Dancers
11 2424 1⋅24=241\cdot 24=24
22 1212 2⋅12=242\cdot 12=24
33 88 3⋅8=243\cdot 8=24
44 66 4⋅6=244\cdot 6=24
66 44 6⋅4=246\cdot 4=24
88 33 8⋅3=248\cdot 3=24
1212 22 12⋅2=2412\cdot 2=24
2424 11 24⋅1=2424\cdot 1=24
What patterns do you see in the table above? Did you notice that the number of groups times the number of dancers per group is always 24?24? This makes sense, since there are always 2424 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 2424, which are:

1,2,3,4,6,8,12,241,2,3,4,6,8,12,24

We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with 11. 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.

Find all the factors of a counting number

  1. 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.
  2. List all the factor pairs.
  3. Write all the factors in order from smallest to largest.
 

example

Find all the factors of 7272. Solution: Divide 7272 by each of the counting numbers starting with 11. 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 The next line would have a divisor of 99 and a quotient of 88. 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 721,2,3,4,6,8,9,12,18,24,36,\text{ and }72
  In the following video we show how to find all the factors of 3030. https://youtu.be/3EL3VA2v9iI

Identify Prime and Composite Numbers

Some numbers, like 7272, have many factors. Other numbers, such as 77, have only two factors: 11 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 11 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 11 whose only factors are 11 and itself. A composite number is a counting number that is not prime.
The table below lists the counting numbers from 22 through 2020 along with their factors. The highlighted numbers are prime, since each has only two factors. Factors of the counting numbers from 22 through 2020, with prime numbers highlighted This figure shows a table with twenty rows and three columns. The first row is a header row. It labels the columns as The prime numbers less than 2020 are 2,3,5,7,11,13,17,and 192,3,5,7,11,13,17,\text{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 11 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 22 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.
 

example

Identify each number as prime or composite:
  1. 8383
  2. 7777

Answer: Solution: 1. Test each prime, in order, to see if it is a factor of 8383 , starting with 22, as shown. We will stop when the quotient is smaller than the divisor.

Prime Test Factor of 83?83?
22 Last digit of 8383 is not 0,2,4,6,or 80,2,4,6,\text{or }8. No.
33 8+3=118+3=11, and 1111 is not divisible by 33. No.
55 The last digit of 8383 is not 55 or 00. No.
77 83÷7=(11.857… )83\div 7=(11.857\dots) No.
1111 83÷11=(7.545… )83\div 11=(7.545\dots) No.
We can stop when we get to 1111 because the quotient (7.545… )(7.545\dots) is less than the divisor. We did not find any prime numbers that are factors of 8383, so we know 8383 is prime. 2. Test each prime, in order, to see if it is a factor of 7777.
Prime Test Factor of 77?77?
22 Last digit is not 0,2,4,6,or 80,2,4,6,\text{or }8. No.
33 7+7=147+7=14, and 1414 is not divisible by 33. No.
55 the last digit is not 55 or 00. No.
77 77÷11=777\div 11=7 Yes.
Since 7777 is divisible by 77, we know it is not a prime number. It is composite.

 

try it

[ohm_question]145441[/ohm_question]
In the following video we show more examples of how to determine whether a number is prime or composite. https://youtu.be/8v7baCT33xw

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