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 · 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.

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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

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

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