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# If a and b are positive integers divisible by 6, is 6 the

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If a and b are positive integers divisible by 6, is 6 the [#permalink]

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02 Sep 2010, 22:28
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If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

(2) a = 3b

What is the fastest (perhaps, algebra?) means of solving the question besides
random plugging numbers under test condition?
[Reveal] Spoiler: OA

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Re: If a and b are positive integers [#permalink]

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02 Sep 2010, 22:41
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note that a=3b does not mean that 6 will be the gcd. example 6, 18 is 6, but 12, 36 is 12 not suff

for a =2b+6= 2(b+3); take b=6,a=18, gcd =6 ; take b=12,a=30; gcd=6; take b=18,a=42; gcd=6 suff

for an algebraic proof: a=6r, b=6s
2. a=3b means 6r=18s or r=3s; a=6s b=18s; cannot conclude about gcd
a=2b+6 means r=2s+1; a=6(2s+1); b=6s no common factors; you can conclude that 6 will be gcd.
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Re: If a and b are positive integers [#permalink]

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03 Sep 2010, 05:20
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gmatbull wrote:
If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

(2) a = 3b

What is the fastest (perhaps, algebra?) means of solving the question besides
random plugging numbers under test condition?

If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

Given: $$a=6x$$ and $$b=6y$$. Question: is $$GCD(a,b)=6$$? Now, If $$x$$ and $$y$$ share any common factor >1then $$GCD(a,b)$$ will be more than 6 if not then $$GCD(a,b)$$ will be 6.

(1) $$a=2b+6$$ --> $$6x=2*6y+6$$ --> $$x=2y+1$$ --> $$x$$ and $$y$$ do not share any factor >1, as if they were we would be able to factor out if from $$2y+1$$. Sufficient.

(2) $$a=3b$$ --> clearly insufficient.

There is also a general rule: if $$a$$ and $$b$$ are multiples of $$k$$ and are $$k$$ units apart from each other then $$k$$ is greatest common divisor of $$a$$ and $$b$$.

For example if $$a$$ and $$b$$ are multiples of 7 and $$a=b+7$$ then 7 is GCD of $$a$$ and $$b$$.

So if we apply this rule to (1) $$a=2b+6$$ --> both $$a$$ and $$2b$$ are multiples of 6 and are 6 apart, so GCD of $$a$$ and $$2b$$ is 6, hence GCD of $$a$$ and $$b$$ is also 6. Sufficient.

Hope it helps.
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Re: If a and b are positive integers [#permalink]

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03 Sep 2010, 10:57
Hey Bunuel....thanx for the rule stated...
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Re: If a and b are positive integers [#permalink]

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03 Sep 2010, 11:04
My appreciation to you guys for taking your time to provide this
informative response; truly grateful.

@Bunuel,
x and y do not share any further factors ( x=2y+1 - consecutives), GCD(a,b) = 6.
Also, thanks for the distance apart info.

Regards to both of you- Bunuel and Mainhoon.
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Re: If a and b are positive integers [#permalink]

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21 Sep 2010, 06:05
Thanks for the solution

Bunuel, may be you should post a collection of all such rules.
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Re: If a and b are positive integers [#permalink]

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16 Feb 2011, 23:26
Bunuel wrote:
gmatbull wrote:
If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

(2) a = 3b

What is the fastest (perhaps, algebra?) means of solving the question besides
random plugging numbers under test condition?

If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

Given: $$a=6x$$ and $$b=6y$$. Question: is $$GCD(a,b)=6$$? Now, If $$x$$ and $$y$$ share any common factor >1then $$GCD(a,b)$$ will be more than 6 if not then $$GCD(a,b)$$ will be 6.

(1) $$a=2b+6$$ --> $$6x=2*6y+6$$ --> $$x=2y+1$$ --> $$x$$ and $$y$$ do not share any factor >1, as if they were we would be able to factor out if from $$2y+1$$. Sufficient.

(2) $$a=3b$$ --> clearly insufficient.

There is also a general rule: if $$a$$ and $$b$$ are multiples of $$k$$ and are $$k$$ units apart from each other then $$k$$ is greatest common divisor of $$a$$ and $$b$$.

For example if $$a$$ and $$b$$ are multiples of 7 and $$a=b+7$$ then 7 is GCD of $$a$$ and $$b$$.

So if we apply this rule to (1) $$a=2b+6$$ --> both $$a$$ and $$2b$$ are multiples of 6 and are 6 apart, so GCD of $$a$$ and $$2b$$ is 6, hence GCD of $$a$$ and $$b$$ is also 6. Sufficient.

Hope it helps.

Thanks for the rule. I picked A as I knew there is "some" rule for common multiples a an integer with that integer as the difference between the common multiples but was not recollecting it..
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Re: If a and b are positive integers [#permalink]

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16 Feb 2011, 23:32
144144 wrote:
these considered to be hard questions?

Definitely above 600 level.
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Re: If a and b are positive integers [#permalink]

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17 Feb 2011, 17:33
Very good approach to alternative 1.

Bunuel wrote:
gmatbull wrote:
If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

(2) a = 3b

What is the fastest (perhaps, algebra?) means of solving the question besides
random plugging numbers under test condition?

If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

Given: $$a=6x$$ and $$b=6y$$. Question: is $$GCD(a,b)=6$$? Now, If $$x$$ and $$y$$ share any common factor >1then $$GCD(a,b)$$ will be more than 6 if not then $$GCD(a,b)$$ will be 6.

(1) $$a=2b+6$$ --> $$6x=2*6y+6$$ --> $$x=2y+1$$ --> $$x$$ and $$y$$ do not share any factor >1, as if they were we would be able to factor out if from $$2y+1$$. Sufficient.

(2) $$a=3b$$ --> clearly insufficient.

There is also a general rule: if $$a$$ and $$b$$ are multiples of $$k$$ and are $$k$$ units apart from each other then $$k$$ is greatest common divisor of $$a$$ and $$b$$.

For example if $$a$$ and $$b$$ are multiples of 7 and $$a=b+7$$ then 7 is GCD of $$a$$ and $$b$$.

So if we apply this rule to (1) $$a=2b+6$$ --> both $$a$$ and $$2b$$ are multiples of 6 and are 6 apart, so GCD of $$a$$ and $$2b$$ is 6, hence GCD of $$a$$ and $$b$$ is also 6. Sufficient.

Hope it helps.
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Re: If a and b are positive integers [#permalink]

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05 Feb 2012, 01:43
Bunuel wrote:
gmatbull wrote:
If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

(2) a = 3b

What is the fastest (perhaps, algebra?) means of solving the question besides
random plugging numbers under test condition?

If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

Given: $$a=6x$$ and $$b=6y$$. Question: is $$GCD(a,b)=6$$? Now, If $$x$$ and $$y$$ share any common factor >1then $$GCD(a,b)$$ will be more than 6 if not then $$GCD(a,b)$$ will be 6.

(1) $$a=2b+6$$ --> $$6x=2*6y+6$$ --> $$x=2y+1$$ --> $$x$$ and $$y$$ do not share any factor >1, as if they were we would be able to factor out if from $$2y+1$$. Sufficient.

(2) $$a=3b$$ --> clearly insufficient.

There is also a general rule: if $$a$$ and $$b$$ are multiples of $$k$$ and are $$k$$ units apart from each other then $$k$$ is greatest common divisor of $$a$$ and $$b$$.

For example if $$a$$ and $$b$$ are multiples of 7 and $$a=b+7$$ then 7 is GCD of $$a$$ and $$b$$.

So if we apply this rule to (1) $$a=2b+6$$ --> both $$a$$ and $$2b$$ are multiples of 6 and are 6 apart, so GCD of $$a$$ and $$2b$$ is 6, hence GCD of $$a$$ and $$b$$ is also 6. Sufficient.

Hope it helps.

But if a and b are both divisible of 6, means that both are even, therefore at least both of them should be divisible by 2.... I am right????? I do not understand why (1) is valid... thanks!!
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Re: If a and b are positive integers [#permalink]

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05 Feb 2012, 02:02
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Saurajm wrote:
But if a and b are both divisible of 6, means that both are even, therefore at least both of them should be divisible by 2.... I am right????? I do not understand why (1) is valid... thanks!!

Yes, both are divisible by 6, which means that they are divisible by 2 and 3.

Next, we have that $$a=6x$$ and $$b=6y$$.

Consider two cases:
1. $$x$$ and $$y$$ share some common factor >1: for example $$x=2$$ and $$y=4$$ then $$a=12$$ and $$b=24$$ --> $$GCD(a,b)=12>6$$;
2. $$x$$ and $$y$$ DO NOT share any common factor >1: for example $$x=5$$ and $$y=2$$ then $$a=30$$ and $$b=12$$ --> $$GCD(a,b)=6$$.

From (1) we have that --> $$x=2y+1$$ --> $$x$$ is one more than multiple of $$y$$. For example: $$x=3$$ and $$y=1$$ OR $$x=5$$ and $$y=2$$ OR $$x=7$$ and $$y=3$$ ... as you can see in all these cases x and y do not share any common factor more than 1. Now, as we concluded above if $$x$$ and $$y$$ DO NOT share any common factor >1, then $$GCD(a,b)=6$$.

Or another way: $$b=6y$$ and $$a=6(2y+1)$$. $$2y$$ and $$2y+1$$ are consecutive integers and consecutive integers do not share any common factor 1. As $$2y$$ has all factors of $$y$$ then $$y$$ and $$2y+1$$ also do not share any common factor but 1, which means that 6 must GCD of $$a$$ and $$b$$

Similar questions to practice:
what-is-the-greatest-common-factor-of-x-and-y-1-x-and-y-are-109273.html
x-and-y-are-positive-integers-such-that-x-8y-12-what-is-the-126743.html

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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]

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22 Feb 2012, 08:25
gmatbull wrote:
If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

(2) a = 3b

I suppose my doubt was not conveyed properly.

By contradiction I meant for a value of b, we get two different a(s) by two statements.

Or else, (18,6) is the only solution possible.
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]

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22 Feb 2012, 08:29
gmatbull wrote:
If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

(2) a = 3b

I suppose my doubt was not conveyed properly.

By contradiction I meant for a value of b, we get two different a(s) by two statements.

Or else, (18,6) is the only solution possible.

I'm not sure I understand what you mean.

Anyway from: a=2b+6 and a=3b there is only one solution possible: a=18 and b=6 (you should just solve the system of equations).
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]

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14 Mar 2012, 03:15
Hi Bunuel, Can you explain the following rule with few examples.

Given: and . Question: is ? Now, If x and y share any common factor >1then GCD (a,b) will be more than 6 if not then GCD (a,b) will be 6.
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]

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14 Mar 2012, 08:08
pavanpuneet wrote:
Hi Bunuel, Can you explain the following rule with few examples.

Given: and . Question: is ? Now, If x and y share any common factor >1then GCD (a,b) will be more than 6 if not then GCD (a,b) will be 6.

Sure. Both $$a$$ and $$b$$ are multiples of 6 --> $$a=6x$$ and $$b=6y$$. Consider two cases:

A. $$x$$ and $$y$$ do not share any common factor >1, for example $$a=6*2=12$$ and $$b=6*3=18$$ --> GCD(a,b)=6. As you can see 2 and 3 did not contribute any common factor to the GCD;

B. $$x$$ and $$y$$ share some common factor >1, for example $$a=6*2=12$$ and $$b=6*4=24$$ --> GCD(a,b)=12, here 2 and 4 contributes common factor 2 to the GCD.

Hope it's clear.
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]

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16 Mar 2012, 00:43
Thank you once again for the prompt response. It's clear.
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]

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26 Jun 2013, 00:39
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]

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26 Jun 2013, 01:19
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gmatbull wrote:
If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

(2) a = 3b

What is the fastest (perhaps, algebra?) means of solving the question besides
random plugging numbers under test condition?

Given a and b are positive integers divisible by 6

From st 1, we have a = 2b+ 6
b is of the form b= 6c where c is a positive integer

therefore we have a = 2*6c+6 ----> a= 6 (2c+1)
b= 6c

Now HCF of a and b will be 6 as
a= 2*3*(2c+1)
b=2*3
Only common factor is 6

So, Ans is A alone sufficient. We can remove options B, C and E as possible answers

Not st 2 says a= 3b
taking b = 6c we have a=18c

a= 2*3^2*c
b=2*3 c
HCF will be 2*3*c that is 6c

Now if c=1 then 6 is a the HCF of both a and b
but if c=2 then 12 is HCF
Since 2 ans choices are possible.

Ans will be A to this Question
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]

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26 Jun 2013, 07:14
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gmatbull wrote:
If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

(2) a = 3b

What is the fastest (perhaps, algebra?) means of solving the question besides
random plugging numbers under test condition?

Question:
6 * x =a and 6 * y =b
we need to find if 6 is the GCD? YES or NO question.
so, basically if we can find a single common factor in x and y, thats it its not a GCD, or if we cant find one then that should also work for us.

(1) a = 2b + 6

6x=12y + 6

x = 2y +1 =>No matter what you do, this will always result in an no common factor.

Thus 6 is the only GCD =>Sufficient

(2) a = 3b

6x = 18y
x=3y

Take y=8, and Y=2 =>This is clearly Not sufficient.

Ans: A
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]

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30 Dec 2013, 09:28
gmatbull wrote:
If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

(2) a = 3b

What is the fastest (perhaps, algebra?) means of solving the question besides
random plugging numbers under test condition?

I think that the best approach is a combination of both

Check this out

We are told that a = 6k , b = 6m

Is gcf > 6?

Statement 1

A = 2b + 6

Replacing we get

6k = 12m + 6

Now if m = 1, k = 3
m = 2, k = 5

As you see there will not be any integers m,k with GCF>1

Hence sufficient

Statement 2

a=3b

6k=18m
k=6m

When m=1, k=6
When m=2, k=12

BINGO, m and k share common factor 2

So replacing we get that 12 is also a factor of both 'a' and 'b'

Then we have to possible answers and thus the statement is not sufficient

Hope it helps
Cheers!
J
Re: If a and b are positive integers divisible by 6, is 6 the   [#permalink] 30 Dec 2013, 09:28

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