If x and y are positive integers, what is the GCF of x and y

Question

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

Bunuel · Accepted Answer

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 co-prime, 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 --> (x-y)^2=1 --> |x-y|=1 --> x and y are consecutive integers --> the greatest common factor of x and y is 1. Sufficient.

Answer: D.

Similar questions to practice:
what-is-the-greatest-common-factor-of-positive-integers-a-126637.html
what-is-the-greatest-common-factor-of-x-and-y-109273.html
what-is-the-greatest-common-divisor-of-positive-integers-m-129802.html
x-and-y-are-positive-integers-such-that-x-8y-12-what-is-the-126743.html
gcd-of-a-b-126427.html
if-a-and-b-are-positive-integers-divisible-by-6-is-6-the-100324.html
if-a-and-b-are-positive-itegers-what-is-the-value-of-a-b-135199.html
is-x-1-a-factor-of-100740.html
find-the-number-that-divides-103251.html
if-n-is-the-least-of-3-consecutive-positive-integers-and-128102.html

JJ2014 · Answer

sorry, can you please elaborate on the red part? how did you solve (x-y)^2 to get |x-y|=1?

Bunuel · Answer

Take the square root from (x-y)^2=1 to get |x-y|=1.

cvaditya · Answer

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.

hikaps14 · Answer

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 ?

Rajkiranmareedu · Answer

|x-y|=1 = Does GCF can be -1.

mau5 · Answer

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.

stonecold · Answer

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

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If x and y are positive integers, what is the GCF of x and y

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

If x and y are positive integers, what is the GCF of x and y

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards