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If x and y are positive integers, what is the GCF of x and y [#permalink]
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15 Dec 2012, 10:30
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If x and y are positive integers, what is the GCF of x and y? (1) When x is divided by y, the remainder is 1. (2) x^2  2xy + y^2 = 1
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Re: If x and y are positive integers, what is the GCF of x and y [#permalink]
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If x and y are positive integers, what is the GCF of x and y?(1) When x is divided by y, the remainder is 1 > x is one more than some multiple of y (x=yq+1). This implies that x and that multiple are consecutive integer. Now, two consecutive integers are coprime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, we have that x and yq do not share any common factor but 1, thus x and y also do not share any common factor but 1, which means that the greatest common factor of x and y is 1. Sufficient. (2) x^2  2xy + y^2 = 1 > (xy)^2=1 > xy=1 > x and y are consecutive integers > the greatest common factor of x and y is 1. Sufficient. Answer: D. Similar questions to practice: whatisthegreatestcommonfactorofpositiveintegersa126637.htmlwhatisthegreatestcommonfactorofxandy109273.htmlwhatisthegreatestcommondivisorofpositiveintegersm129802.htmlxandyarepositiveintegerssuchthatx8y12whatisthe126743.htmlgcdofab126427.htmlifaandbarepositiveintegersdivisibleby6is6the100324.htmlifaandbarepositiveitegerswhatisthevalueofab135199.htmlisx1afactorof100740.htmlfindthenumberthatdivides103251.htmlifnistheleastof3consecutivepositiveintegersand128102.html
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Re: If x and y are positive integers, what is the GCF of x and y [#permalink]
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16 Dec 2012, 08:54
Bunuel wrote: If x and y are positive integers, what is the GCF of x and y?(1) When x is divided by y, the remainder is 1 > x is one more than some multiple of y (x=yq+1). This implies that x and that multiple are consecutive integer. Now, two consecutive integers are coprime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, we have that x and yq do not share any common factor but 1, thus x and y also do not share any common factor but 1, which means that the greatest common factor of x and y is 1. Sufficient. (2) x^2  2xy + y^2 = 1 > (xy)^2=1 > xy=1 > x and y are consecutive integers > the greatest common factor of x and y is 1. Sufficient. Answer: D. Similar questions to practice: whatisthegreatestcommonfactorofpositiveintegersa126637.htmlwhatisthegreatestcommonfactorofxandy109273.htmlwhatisthegreatestcommondivisorofpositiveintegersm129802.htmlxandyarepositiveintegerssuchthatx8y12whatisthe126743.htmlgcdofab126427.htmlifaandbarepositiveintegersdivisibleby6is6the100324.htmlifaandbarepositiveitegerswhatisthevalueofab135199.htmlisx1afactorof100740.htmlfindthenumberthatdivides103251.htmlifnistheleastof3consecutivepositiveintegersand128102.html sorry, can you please elaborate on the red part? how did you solve (xy)^2 to get xy=1?
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Re: If x and y are positive integers, what is the GCF of x and y [#permalink]
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16 Dec 2012, 08:57
JJ2014 wrote: Bunuel wrote: If x and y are positive integers, what is the GCF of x and y?(1) When x is divided by y, the remainder is 1 > x is one more than some multiple of y (x=yq+1). This implies that x and that multiple are consecutive integer. Now, two consecutive integers are coprime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, we have that x and yq do not share any common factor but 1, thus x and y also do not share any common factor but 1, which means that the greatest common factor of x and y is 1. Sufficient. (2) x^2  2xy + y^2 = 1 > (xy)^2=1 > xy=1 > x and y are consecutive integers > the greatest common factor of x and y is 1. Sufficient. Answer: D. Similar questions to practice: whatisthegreatestcommonfactorofpositiveintegersa126637.htmlwhatisthegreatestcommonfactorofxandy109273.htmlwhatisthegreatestcommondivisorofpositiveintegersm129802.htmlxandyarepositiveintegerssuchthatx8y12whatisthe126743.htmlgcdofab126427.htmlifaandbarepositiveintegersdivisibleby6is6the100324.htmlifaandbarepositiveitegerswhatisthevalueofab135199.htmlisx1afactorof100740.htmlfindthenumberthatdivides103251.htmlifnistheleastof3consecutivepositiveintegersand128102.html sorry, can you please elaborate on the red part? how did you solve (xy)^2 to get xy=1? Take the square root from (xy)^2=1 to get xy=1.
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Re: If x and y are positive integers, what is the GCF of x and y [#permalink]
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03 Apr 2013, 20:37
I understand why the answer is D. However, in the first statement, the number x and y dont need to be consecutive numbers to get a remainder 1. for eg 4,9. The GCF is still 1 though.



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Re: If x and y are positive integers, what is the GCF of x and y [#permalink]
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04 Apr 2013, 09:42
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cvaditya wrote: I understand why the answer is D. However, in the first statement, the number x and y dont need to be consecutive numbers to get a remainder 1. for eg 4,9. The GCF is still 1 though. You are correct but what bunuel said was also correct. He stated x & ky are consecutive integers not x & y. in this case 9 & 8 (multiple of 4). 8 & 9 have GCF 1 then 4 & 9 can not have GCF greater than 1. Does this help clarify ?



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Re: If x and y are positive integers, what is the GCF of x and y [#permalink]
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01 May 2013, 06:46
xy=1 = Does GCF can be 1.



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Re: If x and y are positive integers, what is the GCF of x and y [#permalink]
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01 May 2013, 07:12
Rajkiranmareedu wrote: xy=1 = Does GCF can be 1. By definition, HCF of two or more integers is the largest positive integer which divides them without any remainder. For example, HCF of 4 and 2 is +2. Nonetheless, in the given problem, it is mentioned that both x and y are positive. Thus, the GCF IS anyways positive.
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Re: If x and y are positive integers, what is the GCF of x and y [#permalink]
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16 Mar 2016, 09:18
Tremendous Question Here We can see that statement 2 is off course sufficient as two consecutive integers will be co primes statement 1 is sufficient one number is a multiple and the other is a non multiple of y hence no factors in common. Hence D
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Re: If x and y are positive integers, what is the GCF of x and y [#permalink]
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