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

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

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?
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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!

Bunuel please help with this question!!!

Bunuel or any other expert can you please provide a little more explanation with this question.

I do not agree that every integer has to bo consecutive integer as per first statement. We can have 25/6 that also yields remainder 1 or as pointed out 25/24 that yields remainder 1.

statement 1 and 2 tell us the same info that x and y are consecutive integers
if x and y are consecutive then the hcf of both the integers would always be 1
hence D

Not an expert, but I think the answer to this is that even though the remainder is 1 for both 25/6 and 25/24, the GCF will be the same because of how 6 and 24 share prime factors... Prime factors of 24 are 3*2^3 and the prime factors of 6 are 3*2.

6 has even less prime factors than 24 does, so if the GCF of 24 and 25 is 1, then the GCF of 25 and 6 will also be 1.

Hello - so how do we know that the form x = y + 1 means x and y are consecutive integers?

Is there a rule I am missing or a way of thinking about it?

Please help, thanks!

There is a concept called the “Difference Concept” (I don’t know how else to explain it) when you are trying to find the GCF of 2 Numbers (or several)

The GCF must be a factor of the difference between the 2 Numbers and a factor of one of those numbers.


If you think about it as “Gaps” or “jumps” for whatever the positive value of Y is and draw it on a number line, it might make more sense.

Karishma at Veritas has a good blog post on this topic. I believe you can find it in the ultimate quant thread.

For every multiple of Y, whatever that value of Y is, in order to get a remainder of 1——-X will always have to be +1 more than the multiple of Y.


If y = 2———-to get a remainder of 1——, X = 3, 5, 7, 9, 11, etc

Any multiple of Y = 2 will always be 1 shy from the corresponding X


If y = 3 ———-to get a Rem of 1——-X = 7 , 10, 13, 16

Any multiple of Y = 3 will always be 1 shy from the corresponding X


Because the only common factors that Y and X will share has to be a factor of the difference between X and Y———and X will always be + 1 more than a Multiple of Y’s Value————-the only factor of the difference between the 2 values they can possibly share is 1.


I’ll try paging VeritasKarishma over.

I believe she does a great job of explaining the concept. This isn’t turning out as well as I expected. Apologies.

Posted from my mobile device

Fdambro294:

Glad you thought of that post in this question. It can simplify this question a whole lot.
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/1 ... -concepts/

(1) When x is divided by y, the remainder is 1.

This means x is 1 more than a multiple of y. This multiple of y will have all factors of y included. So the next number, x, will have none of those factors. Hence GCF of x and y must be 1.

e.g. say y = 6 (factors are 2 and 3)
Say x = 25 (1 more than 24, a multiple of 6)
24 will have 2 and 3 as factors. So 25 cannot. So the only common factor that 6 and 25 can have is 1.


(2) x^2 – 2xy + y^2 = 1
(x - y)^2 = 1
|x - y| = 1
If difference between x and y is 1, they can have no common factors other than 1.

Answer (D)

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