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# If both x and y are positive integers that are divisible by

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If both x and y are positive integers that are divisible by [#permalink]

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26 Aug 2011, 02:16
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If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

(1) x = 4y
(2) x - 3y = 10
[Reveal] Spoiler: OA

Last edited by Bunuel on 17 Sep 2012, 08:24, edited 1 time in total.
Edited the question.
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Re: Is 10 the greatest common factor of x and y? [#permalink]

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26 Aug 2011, 02:37
If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

(1) x = 4y
(2) x - 3y = 10

1. x=4y tells us x is multiple of 4 not we not sure about Y, but both divisible by 10-Not sufficient
2 x-3y=10, means x is 10 number greater than y but it can be if x 130 y can be 120 but not sure Not Sufficient
1010-1000, 210-200 not sufficient

Now combining both we get Y=10 & x=40 which means 10 is HCF

Remember there is difference between LCM & HCF
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Re: Is 10 the greatest common factor of x and y? [#permalink]

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26 Aug 2011, 02:55
jagdeepsingh1983 wrote:
If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

(1) x = 4y
(2) x - 3y = 10

1. x=4y tells us x is multiple of 4 not we not sure about Y, but both divisible by 10-Not sufficient
2 x-3y=10, means x is 10 number greater than y but it can be if x 130 y can be 120 but not sure Not Sufficient
1010-1000, 210-200 not sufficient

Now combining both we get Y=10 & x=40 which means 10 is HCF

Remember there is difference between LCM & HCF

I think you forgot the "3 times" in (2)
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Re: Is 10 the greatest common factor of x and y? [#permalink]

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27 Aug 2011, 11:22
Berbatov wrote:
If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

(1) x = 4y
(2) x - 3y = 10

Reason -

1. x = 4y
if x = 40, y =10 -> GCF = 10
if x = 80, y =20 -> GCF = 20 ----- not sufficient.

(2) x - 3y = 10
if y = 10, y= 40 -> GCF = 10
if y = 20, y= 70 -> GCF = 10
if y = 30, y= 100 -> GCF = 10 ------- Sufficient.

Hope it helps.!
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Re: Is 10 the greatest common factor of x and y? [#permalink]

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27 Aug 2011, 11:42
Easiest way to do this is to pick numbers.

Berbatov wrote:
If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

(1) x = 4y
(2) x - 3y = 10

Stmt1: x=4y
y=10
x-40
10 is the GCF.
y=100
x=400
100 is the GCF. Yes and No. Not sufficient.

Stmt2: x-3y=10
x=3y+10
y=10
x=40
GCF=10
y=100
x=310
GCF=10
y=50
x=160
GCF=10 Sufficient.
OA B.
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Re: Is 10 the greatest common factor of x and y? [#permalink]

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03 Nov 2011, 10:12
(1) Putting values, we can have 40,10 or 400,100. GCD can be 10 or 100. NOT SUFFICIENT.
(2) x - 3y = 10, again putting values, we can have only 40,10 or 70,20.
Now let us substitute x = 10a and y = 10b.
10a – 30b = 10,
Thus a = (10 +30b) / 10
We can conclude for all values of b, 10 + 30b must be a multiple of 10, hence GCD of x and y is indeed 10.
SUFFICIENT.
Hence ans is B
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Re: If both x and y are positive integers that are divisible by [#permalink]

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16 Sep 2012, 05:43
Is there any way other than substituting values?
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Re: If both x and y are positive integers that are divisible by [#permalink]

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16 Sep 2012, 05:54
Berbatov wrote:
If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

(1) x = 4y
(2) x - 3y = 10

If 10 is the greatest common divisor of $$x$$ and $$y$$ then $$x=10a, \, y=10b$$ for some positive integers $$a$$ and $$b$$, where $$a$$ and $$b$$ are co-prime (meaning their greatest common divisor is 1).

(1) Obviously not sufficient. It just states that $$x$$ is 4 times bigger than $$y$$, which can hold even without the two numbers being divisible by 10.
For example $$x=4,\,y=1.$$

(2) Again, not sufficient. $$x=3y+10$$, $$x$$ and $$y$$ not necessarily divisible by 10. For example $$x=13,\,y=1.$$

(1) and (2) together: solving the two equations we obtain unique values for $$x$$ and $$y$$: $$x=40,\,y=10.$$
Sufficient.

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Re: If both x and y are positive integers that are divisible by [#permalink]

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16 Sep 2012, 07:01
Guess they have mentioned in the question that x & y are divisible by 10. So, x & y cant take values 13 and 1. But still I feel 2nd statement is not sufficient.
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Re: If both x and y are positive integers that are divisible by [#permalink]

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16 Sep 2012, 07:39
sravs27 wrote:
Guess they have mentioned in the question that x & y are divisible by 10. So, x & y cant take values 13 and 1. But still I feel 2nd statement is not sufficient.

Thanks.
I am such an astronaut...

So, let's try again:

Assume $$x=10a$$ and $$y=10b$$, for some positive integers $$a$$ and $$b.$$
$$x$$ and $$y$$ have 10 as their greatest common divisor if and only if $$a$$ and $$b$$ are co-prime (their greatest common divisor is 1).

(1) $$x=4y$$ translates into $$10a=40$$b or $$a=4b$$.
$$a$$ and $$b$$ can be co-prime only if $$b=1.$$
Not sufficient.

(2) $$x-3y=10$$ becomes $$10a-30b = 10$$ or $$a-3b=1$$.
If $$a$$ and $$b$$ have a common divisor $$d$$ (some positive integer), then $$a=md$$ and $$b=nd$$ for some positive integers $$m$$ and $$n.$$
It follows that $$dm-3dn = d(m-3n)=1$$, which means that $$d$$ must be a divisor of 1, so $$d=1.$$
It means that $$a$$ and $$b$$ are co-prime.
Sufficient.

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Re: If both x and y are positive integers that are divisible by [#permalink]

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17 Sep 2012, 08:33
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If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

Given: $$x=10m$$ and $$y=10n$$, for some positive integers $$m$$ and $$n$$.

(1) x = 4y. If $$x=40$$ and $$y=10$$, then the answer is YES but $$x=80$$ and $$y=20$$, then the answer is NO. Not sufficient.

(2) x - 3y = 10 --> $$10m-3*(10n)=10$$ --> $$m-3n=1$$ --> $$m=3n+1$$. $$m$$ and $$3n$$ are consecutive integers. Any two consecutive positive integers are co-prime, which means that they do not share any common factor but 1. For example: 3 and 4, 5 and 6, 100 and 101, are consecutive integers and they do not share any common factor bu 1. So, $$m$$ and $$3n$$ do not share any common factors but 1, which means that $$m$$ and $$n$$ also do not share any common factors but 1, therefore the greatest common factor of $$x=10m$$ and $$y=10n$$ is 10. Sufficient.

Hope it's clear.
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Re: If both x and y are positive integers that are divisible by [#permalink]

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13 Sep 2013, 00:14
Bumping for review and further discussion.
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Re: If both x and y are positive integers that are divisible by [#permalink]

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19 Oct 2013, 00:18
Bunuel wrote:
If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

Given: $$x=10m$$ and $$y=10n$$, for some positive integers $$m$$ and $$n$$.

(1) x = 4y. If $$x=40$$ and $$y=10$$, then the answer is YES but $$x=80$$ and $$y=20$$, then the answer is NO. Not sufficient.

(2) x - 3y = 10 --> $$10m-3*(10n)=10$$ --> $$m-3n=1$$ --> $$m=3n+1$$. $$m$$ and $$3n$$ are consecutive integers. Any two consecutive positive integers are co-prime, which means that they do not share any common factor but 1. For example: 3 and 4, 5 and 6, 100 and 101, are consecutive integers and they do not share any common factor bu 1. So, $$m$$ and $$3n$$ do not share any common factors but 1, which means that $$m$$ and $$n$$ also do not share any common factors but 1, therefore the greatest common factor of $$x=10m$$ and $$y=10n$$ is 10. Sufficient.

Hope it's clear.

I am Sorry I can't get it, how are $$m$$ and $$3n$$ consecutive integers?

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Re: If both x and y are positive integers that are divisible by [#permalink]

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19 Oct 2013, 00:51
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suk1234 wrote:
Bunuel wrote:
If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

Given: $$x=10m$$ and $$y=10n$$, for some positive integers $$m$$ and $$n$$.

(1) x = 4y. If $$x=40$$ and $$y=10$$, then the answer is YES but $$x=80$$ and $$y=20$$, then the answer is NO. Not sufficient.

(2) x - 3y = 10 --> $$10m-3*(10n)=10$$ --> $$m-3n=1$$ --> $$m=3n+1$$. $$m$$ and $$3n$$ are consecutive integers. Any two consecutive positive integers are co-prime, which means that they do not share any common factor but 1. For example: 3 and 4, 5 and 6, 100 and 101, are consecutive integers and they do not share any common factor bu 1. So, $$m$$ and $$3n$$ do not share any common factors but 1, which means that $$m$$ and $$n$$ also do not share any common factors but 1, therefore the greatest common factor of $$x=10m$$ and $$y=10n$$ is 10. Sufficient.

Hope it's clear.

I am Sorry I can't get it, how are $$m$$ and $$3n$$ consecutive integers?

AS Bunuel Proved in the second statement m = 3n + 1 that means m is 1 greater than 3n or m and 3n are consecutive.

I hope it is clear now.

I am following GMAT quant since 2 months and I am in awe of how Bunuel goes about each question so thoroughly and perfectly. I wish I may achieve half of that.

Probably "the greatest" Mathematician I have come across all my life

Regards,
Abhinav
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Re: If both x and y are positive integers that are divisible by [#permalink]

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19 Jan 2014, 08:24
Bunuel wrote:
If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

Given: $$x=10m$$ and $$y=10n$$, for some positive integers $$m$$ and $$n$$.

(1) x = 4y. If $$x=40$$ and $$y=10$$, then the answer is YES but $$x=80$$ and $$y=20$$, then the answer is NO. Not sufficient.

(2) x - 3y = 10 --> $$10m-3*(10n)=10$$ --> $$m-3n=1$$ --> $$m=3n+1$$. $$m$$ and $$3n$$ are consecutive integers. Any two consecutive positive integers are co-prime, which means that they do not share any common factor but 1. For example: 3 and 4, 5 and 6, 100 and 101, are consecutive integers and they do not share any common factor bu 1. So, $$m$$ and $$3n$$ do not share any common factors but 1, which means that $$m$$ and $$n$$ also do not share any common factors but 1, therefore the greatest common factor of $$x=10m$$ and $$y=10n$$ is 10. Sufficient.

Hope it's clear.

We can also note on statement 2 that since X and Y are both multiples of 10 and the difference is a multiple of 10 then 10 must be their GCF, if x and y had a higher GCF say like 20,30,40 etc... then the difference of two multiples of 20 will never yield 10 as a difference, same with others

Just my 2cents

Additional note: If the second statement were x-y = 10 we could also know that the GCF cannot exceed 10 because the GCF is always less than the difference between two numbers and exactly 10 when these two numbers are consecutive integers. Therefore we would know that these two numbers x and y are consecutive integers.

But that's for another problem I guess

Hope it helps
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Re: If both x and y are positive integers that are divisible by [#permalink]

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Re: If both x and y are positive integers that are divisible by [#permalink]

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07 Apr 2016, 18:45
Berbatov wrote:
If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

(1) x = 4y
(2) x - 3y = 10

1. x=40, y = 10 -> GCF = 10 or X=200, y=50. GCF=50 -> A and D out.
2. x-3y=10 or
(x-10)/3 = y
so (x-10)/3 is a number divisible by 10.
x=40 - y=10
x=70 - y=20
x=100 - y=30
so as we can see, GCF always is 10. sufficient.
Re: If both x and y are positive integers that are divisible by   [#permalink] 07 Apr 2016, 18:45
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