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

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

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.

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

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

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.

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

Re: If a and b are positive integers [#permalink]
05 Feb 2012, 02:02

1

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Expert's post

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

Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]
14 Mar 2012, 08:08

Expert's post

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.

Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]
26 Jun 2013, 07:14

1

This post received KUDOS

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

_________________

PS: Like my approach? Please Help me with some Kudos.