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# If x and y are positive integers, what is the greatest common factor

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Manager
Joined: 13 Sep 2016
Posts: 120
If x and y are positive integers, what is the greatest common factor  [#permalink]

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25 Oct 2016, 10:36
1
2
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Difficulty:

55% (hard)

Question Stats:

55% (01:42) correct 45% (01:48) wrong based on 145 sessions

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If x and y are positive integers, what is the greatest common factor of x and y?

(1) When x is divided by y, the remainder is 1.
(2) x^2 – 2xy + y^2 = 1
Manager
Joined: 29 Aug 2008
Posts: 110
Re: If x and y are positive integers, what is the greatest common factor  [#permalink]

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25 Oct 2016, 10:55
1
Question tells us that both x and y are positive integers.

Asks us to find the GCF for them:

Statement 1: x is of the form x = ny + 1, so if we keep n =1 it can be written as x = y + 1 which makes them consecutive integers and two consecutive integers have 1 as the common factor. You can try with other values too the GCF would be 1, so statement 1 itself is sufficient.

Statement 2: x^2 – 2xy + y^2 = 1

can be written (x-y)^2 = 1

Taking square root of both the sides we get:

x - y = +1 or -1

which can be written as x = y + 1 (in case of +1), if you notice they are again consecutive integers.

in case of -1 it can be written as x = y -1, same again they are consecutive integers.

In both cases we are getting only 1 as the GCF, so statement 2 is also sufficient.

Hence option D.

HTH
Manager
Joined: 03 Sep 2018
Posts: 57
If x and y are positive integers, what is the greatest common factor  [#permalink]

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03 Jan 2019, 04:34
gauravk

I disagree with statement 1: If

$$\frac{x}{y} = \frac{2*17}{2*4} –> GCF(x,y) = 2, but: remainder = 1$$

but if

$$\frac{x}{y}=\frac{17}{4} –> GCF(x,y)=1, but: remainder = 1$$

Two different answers, hence, insufficient. What am I missing?
_________________

Manager
Joined: 02 Aug 2015
Posts: 155
If x and y are positive integers, what is the greatest common factor  [#permalink]

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03 Jan 2019, 07:53
1
ghnlrug wrote:

I disagree with statement 1: If

$$\frac{x}{y} = \frac{2*17}{2*4} –> GCF(x,y) = 2, but: remainder = 1$$

but if

$$\frac{x}{y}=\frac{17}{4} –> GCF(x,y)=1, but: remainder = 1$$

Two different answers, hence, insufficient. What am I missing?

Hey ghnlrug,

$$\frac{x}{y} = \frac{2*17}{2*4} –> GCF(x,y) = 2, but: remainder = 1$$

The reminder of fraction 34/8 is 2 and not 1. I always suggest not to split the numbers while calculating the reminders.

Cheers!
If x and y are positive integers, what is the greatest common factor   [#permalink] 03 Jan 2019, 07:53
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