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Definitions: Relatively Prime, Proper Factor

Date: 9/11/96 at 18:21:42
From: F. Fitzpatrick
Subject: Definitions: Relatively Prime, Proper Factor

What does it mean to be relatively prime?
What is a proper factor?

Date: 9/11/96 at 19:18:36
From: Doctor Tom
Subject: Re: Definitions: Relatively Prime, Proper Factor

Two integers are relatively prime if the largest divisor they
have in common is one.

For example, 8 and 12 are NOT relatively prime, because the
number 4 divides evenly into both of them.

The numbers 35 and 27 are relatively prime since no number
larger than 1 goes evenly into both.

To find out if two numbers are relatively prime, you can use
"Euclid's Algorithm" to find their greatest common divisor (GCD).
If the GCD is equal to 1, the numbers are relatively prime.

Look up GCD or Euclid's algorithm in any elementary book on
number theory.

>What is a proper factor?

A proper factor is a factor other than 1 or the number.
[Ed note: see below.]

For example, the numbers that divide evenly into 12 include:  
1, 2, 3, 4, 6, and 12.  They are all factors of 12 but only 
2, 3, 4, and 6 are proper factors.

-Doctor Tom,  The Math Forum
 Check out our web site!   

Date: 9/12/96 at 0:0:4
From: F. Fitzpatrick
Subject: Re: Definitions: Relatively Prime, Proper Factor

Thank you for your response to my question (and so fast!!!).  It 
really helped a lot, and my Algebra teacher is going to think I am a 
genius (well, maybe not).  Thanks again for 
your response to my question.  

Date: 04/29/2001 at 18:52:36
From: Anthony Carpenter
Subject: Can 1 be considered a Proper Factor?

Dr. Math,

I did a search on your Web site to find some information on Proper 
Factors. I understand the concept; however, I am not sure what exactly 
makes up proper factors of a numeral. For example, the proper factors 
of 6 are 1, 2, and 3, is that correct? 

The reason I am asking is that I am taking a Survey of Mathematics 
class and it has some examples where 1 is considered a proper factor.  
Specifically my book says that the proper factors of 8 are 1, 2, and 
4. In a question I found on your site for the proper factors of 12, 
someone said that the proper factors would be 2, 3, 4, and 6. What 
about 1?

Any help you can provide would be appreciated.
Thank you.

Date: 04/30/2001 at 14:10:45
From: Doctor Greenie
Subject: Re: Can 1 be considered a Proper Factor?

Hi, Anthony -

I think you will find there are different opinions regarding the 
correct way to answer.

Clearly, your question cannot be answered until we have an agreement 
on what the terms "factor" and "proper factor" mean, and I will throw 
in the terms "divisor" and "proper divisor" also. I think you will 
find disagreement among mathematicians on the meanings of some of 
these terms.

You shouldn't find any disagreement among mathematicians that the 
prime factorization of 12 is 2*2*3. But to some mathematicians, this 
prime factorization means that 12 "has two prime factors" ("2 and 3"), 
whereas to others it means that 12 "has three prime factors" ("2, 2, 
and 3").

When you talk about "the factors of a number" instead of "the prime 
factors of a number," you are getting into language that is even more 
ambiguous. If you ask 1000 mathematicians to name the factors of 12, I 
suspect a large percentage will interpret the question as meaning 
prime factors, and they will respond, as suggested in the preceding 
paragraph, either "2 and 3" or "2, 2, and 3."

But undoubtedly many of them will interpret the question as meaning 
"divisors" instead of "factors". They might respond with "1, 2, 3, 4, 
6, and 12," if they interpret the question as meaning all divisors, or 
"1, 2, 3, 4, and 6" if they interpret the question as meaning only 
proper divisors. I am quite certain that many mathematicians will 
claim that the only number "whose factors are 1, 2, 3, 4, and 6" is 
1*2*3*4*6 = 144.

Getting back to your specific question as nearly as I can, you should 
find no disagreement among mathematicians that the "proper divisors" 
of 8 are 1, 2, and 4; or that the "proper divisors" of 12 are 1, 2, 3, 
4, and 6. But I'm certain you will find disagreement as to whether the 
"proper factors" of 8 (or 12, or any other number) include the number 

- Doctor Greenie, The Math Forum   

Date: 04/30/2001 at 17:13:27
From: Doctor Peterson
Subject: Re: Can 1 be considered a Proper Factor?

Hi, Anthony.

Your book is right; generally, the word "proper" means "except the 
whole." For example, a proper subset is any subset other than the 
whole set; the empty set would be a proper subset. Similarly, 12 is 
not a proper factor of 12, but 1 is. We need to correct our answer, 

Interestingly, it seems that we are not alone in doing this. In 
searching for other references to see if this error is common, I found 
that this page Glossary   

must have been corrected since Google cached it, because the search 

    Proper Factor

    Factors of a number other than 1 and itself
    Example: Proper Factors of 12 would be 2,3,4,and 6 

while the current page has

    Proper Factor

    Any whole-number factor of a number except the number itself. 
    Example: For example, the factors of 10 are 1, 2, 5, and 10. The
    proper factors of 10 are 1, 2, and 5.

Here is another reference to support the proper definition: - proper factor   

which defines it as

    A factor of a number other than the number itself. Unity is
    considered to be a proper factor.

Of course, you'll find "proper divisor" used more often, as in our 
Number Glossary FAQ.

- Doctor Peterson, The Math Forum   

Date: 02/08/2003 at 18:52:52
From: Jennifer Siegel
Subject: Relatively prime

I am still unclear about relatively prime. 

The question in my book (for an abstract algebra course) says: 
List all the positive integers that are less than 12 and relatively 
prime to 12. I come up with 2, 3, 5, 7, 11, since all combinations of 
pairs of these (or more than two) are all commonly divisible by only 1.
However, my book says there are only four such numbers, but it 
doesn't list them. I cannot determine which one of these numbers to 
eliminate. Any ideas or suggestions would be GREATLY appreciated!

Jennifer Siegel

From: Doctor Jodi
Subject: Re: Relatively prime

Hi Jennifer,

Your textbook should have some examples! Let me give you some.

Are 7 and 14 relatively prime? No, because their GCD is 7. 
Or because 14 and 7 are both divisible by 7.

Are 2 and 30 relatively prime? No, because their GCD is 2. 
Or because 30 and 2 are both divisible by 2.

Are 15 and 35 relatively prime? No, because their GCD is 5. 
Or because 35 and 15 have a common factor: they're both divisible 
by 5.

Does this help answer your question?

So which numbers are relatively prime to 12? (Hint: don't forget 1.)

Write back if you'd like to discuss this some more.

- Doctor Jodi, The Math Forum   
Associated Topics:
High School Number Theory

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