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

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

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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
[Reveal] Spoiler: OA

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

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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 divisible by 6, is 6 the greatest com [#permalink]

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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.

Answer: A.

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 divisible by 6, is 6 the greatest com [#permalink]

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New post 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.

Answer: A.

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 divisible by 6, is 6 the greatest com [#permalink]

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New post 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.

Answer: A.

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 divisible by 6, is 6 the greatest com [#permalink]

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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

Hope it helps.
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Re: If a and b are positive integers divisible by 6, is 6 the greatest com [#permalink]

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New post 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 greatest com [#permalink]

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New post 22 Feb 2012, 08:29
vailad 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



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 greatest com [#permalink]

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New post 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 greatest com [#permalink]

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New post 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 greatest com [#permalink]

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

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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 greatest com [#permalink]

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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 greatest com [#permalink]

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New post 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

Answer is A

Hope it helps
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Re: If a and b are positive integers divisible by 6, is 6 the greatest com [#permalink]

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New post 29 Sep 2014, 13:09
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.

Answer: A.

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.


It would be great if one could find more questions involving GCD/GCF as they seem to be little tricky more often than not.
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Re: If a and b are positive integers divisible by 6, is 6 the greatest com [#permalink]

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earnit wrote:
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.

Answer: A.

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.


It would be great if one could find more questions involving GCD/GCF as they seem to be little tricky more often than not.


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Hope it helps.
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Re: If a and b are positive integers divisible by 6, is 6 the greatest com [#permalink]

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New post 31 Mar 2016, 11:01
Bunuel, a question if you don't mind..

Why do you say that 12 and 18 don't share any common factors >1 ? they do. 12=2*2*3 18=3*3*2, so 2 and 3 are shared. and they form the GCF in this case.

More, if a and b are multiples of 6, and a=2b+6, if b=6, doesn't this mean that a=3b? (or a=18 simply).. if b=12 then a=5b etc. in any case, a and b are NOT 6 apart (or are more than 6 apart) from each other, so we get a NO answer to the question, so Sufficient.

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

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New post 31 Mar 2016, 11:15
iliavko wrote:
Bunuel, a question if you don't mind..

Why do you say that 12 and 18 don't share any common factors >1 ? they do. 12=2*2*3 18=3*3*2, so 2 and 3 are shared. and they form the GCF in this case.

More, if a and b are multiples of 6, and a=2b+6, if b=6, doesn't this mean that a=3b? (or a=18 simply).. if b=12 then a=5b etc. in any case, a and b are NOT 6 apart (or are more than 6 apart) from each other, so we get a NO answer to the question, so Sufficient.

Thank you!


Where did I say that 12 and 18 do not share any common factor?
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Re: If a and b are positive integers divisible by 6, is 6 the greatest com [#permalink]

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New post 31 Mar 2016, 12:16
I am referring to this reply you gave: A) says it.

Bunuel wrote:
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 greatest com [#permalink]

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New post 31 Mar 2016, 12:21
iliavko wrote:
I am referring to this reply you gave: A) says it.

Bunuel wrote:
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.


Thank you!


Please read carefully:\(x\) and \(y\) do not share any common factor >1, NOT a and b.
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