Definitions: Relatively Prime, Proper FactorDate: 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! http://mathforum.org/dr.math/ 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 1. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ 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, above. 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 Allmath.com: Glossary http://www.allmath.com/glossary.asp?page=p must have been corrected since Google cached it, because the search found 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: Cenius.net - proper factor http://www.cenius.fsnet.co.uk/refer/maths/articles/p/properfactor.html 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 http://mathforum.org/dr.math/ 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! Sincerely, 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 http://mathforum.org/dr.math/ |
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