Finding All the Factors of a Number
Learning Outcomes
- Find all the factors of a number
- Determine whether a number is prime or composite
Factors
If [latex]a\cdot b=m[/latex], then [latex]a\text{ and }b[/latex] are factors of [latex]m[/latex], and [latex]m[/latex] is the product of [latex]a\text{ and }b[/latex].Number of Groups | Dancers per Group | Total Dancers |
---|---|---|
[latex]1[/latex] | [latex]24[/latex] | [latex]1\cdot 24=24[/latex] |
[latex]2[/latex] | [latex]12[/latex] | [latex]2\cdot 12=24[/latex] |
[latex]3[/latex] | [latex]8[/latex] | [latex]3\cdot 8=24[/latex] |
[latex]4[/latex] | [latex]6[/latex] | [latex]4\cdot 6=24[/latex] |
[latex]6[/latex] | [latex]4[/latex] | [latex]6\cdot 4=24[/latex] |
[latex]8[/latex] | [latex]3[/latex] | [latex]8\cdot 3=24[/latex] |
[latex]12[/latex] | [latex]2[/latex] | [latex]12\cdot 2=24[/latex] |
[latex]24[/latex] | [latex]1[/latex] | [latex]24\cdot 1=24[/latex] |
[latex]1,2,3,4,6,8,12,24[/latex]
We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with [latex]1[/latex]. 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
- 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.
example
Find all the factors of [latex]72[/latex]. Solution: Divide [latex]72[/latex] by each of the counting numbers starting with [latex]1[/latex]. If the quotient is a whole number, the divisor and quotient are a pair of factors. The next line would have a divisor of [latex]9[/latex] and a quotient of [latex]8[/latex]. 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 [latex]1,2,3,4,6,8,9,12,18,24,36,\text{ and }72[/latex]Identify Prime and Composite Numbers
Some numbers, like [latex]72[/latex], have many factors. Other numbers, such as [latex]7[/latex], have only two factors: [latex]1[/latex] 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 [latex]1[/latex] 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 [latex]1[/latex] whose only factors are [latex]1[/latex] and itself. A composite number is a counting number that is not prime.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 [latex]2[/latex] 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.
example
Identify each number as prime or composite:- [latex]83[/latex]
- [latex]77[/latex]
Answer: Solution: 1. Test each prime, in order, to see if it is a factor of [latex]83[/latex] , starting with [latex]2[/latex], as shown. We will stop when the quotient is smaller than the divisor.
Prime | Test | Factor of [latex]83?[/latex] |
---|---|---|
[latex]2[/latex] | Last digit of [latex]83[/latex] is not [latex]0,2,4,6,\text{or }8[/latex]. | No. |
[latex]3[/latex] | [latex]8+3=11[/latex], and [latex]11[/latex] is not divisible by [latex]3[/latex]. | No. |
[latex]5[/latex] | The last digit of [latex]83[/latex] is not [latex]5[/latex] or [latex]0[/latex]. | No. |
[latex]7[/latex] | [latex]83\div 7=(11.857\dots)[/latex] | No. |
[latex]11[/latex] | [latex]83\div 11=(7.545\dots)[/latex] | No. |
Prime | Test | Factor of [latex]77?[/latex] |
---|---|---|
[latex]2[/latex] | Last digit is not [latex]0,2,4,6,\text{or }8[/latex]. | No. |
[latex]3[/latex] | [latex]7+7=14[/latex], and [latex]14[/latex] is not divisible by [latex]3[/latex]. | No. |
[latex]5[/latex] | the last digit is not [latex]5[/latex] or [latex]0[/latex]. | No. |
[latex]7[/latex] | [latex]77\div 11=7[/latex] | Yes. |
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[ohm_question]145441[/ohm_question]Contribute!
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- Question ID 145441, 145439. Authored by: Lumen Learning. License: CC BY: Attribution. License terms: IMathAS Community License.
CC licensed content, Shared previously
- Ex 1: Determine Factors of a Number. License: CC BY: Attribution.
- Ex 1: Prime Factorization Using Stacked Division. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Determine if Numbers Are Prime or Composite (Algorithm). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
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- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].