Alespiss 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

IMO: D because (1) tells me that 6 is the greatest common divisor, while (2) tells me that 6 isn't, but I can't figure out why the answer is:

[spoiler]OA:A[/spoiler]

Source: MGMAT

Alright, let's try to rewrite the question as a formula...

a mod 6 = 0
b mod 6 = 0
a and b are positive integers

Does GCD(a,b) = 6?

To find the greatest common divisor of two integers, we break down each into the product of prime factors and then find all prime factors that exist in each number's breakdown. So, for 6 to be the GCD of a and b, we need to prove that there is only one 2 and one 3 in common between each number's set of prime factors!

Part (1)
a = 2b + 6

b = x*6, where x represents the product of all prime factors of b (other than on of the 2's and one of the 3's)
2b = 2x*6
2b + 6 = (2x+1)*6 = a

So, we can see that the other factors of a multiplied together are equal to (2*x+1), which means that there is an offset of 1 regardless of which factor of b/6 is divided into a.

Therefore, GCD(a,b) = 6 --> SUFFICIENT

Part (2)
a = 3b

This is easily proven insufficient by clever selection of values for a and b:

a = 6 | b = 18 -> GCD(a,b) = 6
a = 12 | b = 36 -> GCD(a,b) = 12

INSUFFICIENT

A

(1) tells me that 6 is the greatest common divisor, while (2) tells me that 6 isn't

The point in case is that, the two statements in the GMAT will never contradict each other!!

if you arrive at such a situation, be doubly sure!!!

Neo Anderson wrote:

(1) tells me that 6 is the greatest common divisor, while (2) tells me that 6 isn't

The point in case is that, the two statements in the GMAT will never contradict each other!!

if you arrive at such a situation, be doubly sure!!!

You know Kung Fu?

Alespiss 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

IMO: D because (1) tells me that 6 is the greatest common divisor, while (2) tells me that 6 isn't, but I can't figure out why the answer is:

[spoiler]OA:A[/spoiler]

Source: MGMAT

The portion in red is not correct.
The GCF could be 6; in fact, it could be ANY MULTIPLE OF 6.

Statement 2: a = 3b.
Rephrased:
a/b = 3.
Since a/b is an integer, b is a factor of a.
Since b is a factor of a, and the greatest factor of b is b itself, the GCF of a and b is b.
Since b can be any multiple of 6, the GCF of a and b can be any multiple of 6:
If b=6, then a=18, and the GCF=b=6.
If b=12, then a=36, and the GCF=b=12.
If b=18, then a=54, and the GCF=b=18.
And so on.
INSUFFICIENT.

I got it!

Thank you very much for your help

GMATGuruNY wrote:

Alespiss 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

IMO: D because (1) tells me that 6 is the greatest common divisor, while (2) tells me that 6 isn't, but I can't figure out why the answer is:

[spoiler]OA:A[/spoiler]

Source: MGMAT

The portion in red is not correct.
The GCF could be 6; in fact, it could be ANY MULTIPLE OF 6.

Statement 2: a = 3b.
Rephrased:
a/b = 3.
Since a/b is an integer, b is a factor of a.
Since b is a factor of a, and the greatest factor of b is b itself, the GCF of a and b is b.
Since b can be any multiple of 6, the GCF of a and b can be any multiple of 6:
If b=6, then a=18, and the GCF=b=6.
If b=12, then a=36, and the GCF=b=12.
If b=18, then a=54, and the GCF=b=18.
And so on.
INSUFFICIENT.

Could you explain Statement 1 as well? Thanks!

For whatever reason, this random question is in my head and I need some clarification please:

1) 12/6 (a/b) = 2. Hence b is a factor of a

2) 3/6 (3/6) = 0.5. So b is not a factor of a???

In the question it only says that A and B are integers. It does not say that the quotient is an integer.

[quote="GMATGuruNY"][quote="Alespiss"]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

IMO: D because (1) tells me that 6 is the greatest common divisor, [color=red]while (2) tells me that 6 isn't[/color], but I can't figure out why the answer is:

[spoiler]OA:A[/spoiler]

Source: MGMAT[/quote]

The portion in red is not correct.
The GCF [i]could[/i] be 6; in fact, it could be ANY MULTIPLE OF 6.

[b]Statement 2: a = 3b.[/b]
Rephrased:
a/b = 3.
Since a/b is an integer, b is a factor of a.
Since b is a factor of a, and the greatest factor of b is b itself, the GCF of a and b is b.
Since b can be any multiple of 6, the GCF of a and b can be any multiple of 6:
If b=6, then a=18, and the GCF=b=6.
If b=12, then a=36, and the GCF=b=12.
If b=18, then a=54, and the GCF=b=18.
And so on.
INSUFFICIENT.[/quote]

Please explain Statement 1 as well.

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

Could you explain Statement 1 as well? Thanks!

One approach is to plug in values.

Statement 1: a = 2b + 6
If b=6, then a = 2(6) + 6 = 18
The GCD of a=18 and b=6 is 6.

If b=12, then a = 2(12) + 6 = 30.
The GCD of a=30 and b=12 is 6.

One more combination if we want to be really, really safe.
If b=18, then a = 2(18) + 6 = 42.
The GCD of a=42 and b=18 is 6.

In every case, the GCD is 6.
SUFFICIENT.

Here's a proof.

Since a and b are each a multiple of 6, 6 is a factor of both a and b.
Thus, the GCD is at least 6.
The only question is whether a and b could have factors in common other than 6, in which case the GCD would be GREATER than 6.

Since b is a multiple of 6, let b = 6k, where k is an integer.

Statement 1: a = 2b + 6
Substituting b = 6k into a = 2b + 6, we get:
a = 2(6k) + 6
a = 6(2k + 1).

a has the following factors: 6 and 2k+1.
Since b = 6k, b has the following factors: 3 and 2k.
2k and 2k+1 are CONSECUTIVE integers.
Consecutive integers are COPRIMES: they share no factors other than 1.
Since 2k and 2k+1 have no factors in common other than 1, it is not possible for a and b to have a GCD greater than 6.
SUFFICIENT.