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If a and b are positive integers divisible by 6

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If a and b are positive integers divisible by 6

by rsarashi » Wed May 17, 2017 10:13 am
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

OAA

It is given that both a and b are divisible by 6.

1) a = 2b + 6

If I put b=6, then a will be 18, which means 6 is the GCF -- YES

If I put a=6, then b will not be divisible by 6, so that means 6 will not be the GCF -- NO.


So how can A be the answer.

Please explain.

Thanks.
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by [email protected] » Wed May 17, 2017 10:20 am
Hi rsarashi,

When working through GMAT questions, you have to pay careful attention to the information that you're given. Here, we're told that A and B are POSITIVE INTEGERS divisible by 6.

In your second example, you're thinking about A = 6... but that would give you B = 0... That is NOT allowed here (both variables have to be POSITIVE INTEGERS and 0 is not positive). Working 'up' from your initial example, you could try...

B=6, A=18
B=12, A=30
B=18, A=42
B=24, A=54
Etc.

What is the answer to the question when you consider these possibilities....

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by rsarashi » Thu May 18, 2017 8:51 am
When working through GMAT questions, you have to pay careful attention to the information that you're given. Here, we're told that A and B are POSITIVE INTEGERS divisible by 6.

In your second example, you're thinking about A = 6... but that would give you B = 0... That is NOT allowed here (both variables have to be POSITIVE INTEGERS and 0 is not positive). Working 'up' from your initial example, you could try...

B=6, A=18
B=12, A=30
B=18, A=42
B=24, A=54
Etc.

What is the answer to the question when you consider these possibilities....

Thank you so much for your reply. I understood .

Just a quick question, we can not put a=12 because if we put, then b will be 3, and 3 is not divisible by 6. So this case is invalid right?

Please confirm if my understanding is correct.

Thanks.
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by Jay@ManhattanReview » Thu May 18, 2017 11:11 am
rsarashi wrote:
When working through GMAT questions, you have to pay careful attention to the information that you're given. Here, we're told that A and B are POSITIVE INTEGERS divisible by 6.

In your second example, you're thinking about A = 6... but that would give you B = 0... That is NOT allowed here (both variables have to be POSITIVE INTEGERS and 0 is not positive). Working 'up' from your initial example, you could try...

B=6, A=18
B=12, A=30
B=18, A=42
B=24, A=54
Etc.

What is the answer to the question when you consider these possibilities....

Thank you so much for your reply. I understood .

Just a quick question, we can not put a=12 because if we put, then b will be 3, and 3 is not divisible by 6. So this case is invalid right?

Please confirm if my understanding is correct.

Thanks.
Yes, rsarash. That's correct. One must choose examples such that a and b each are divisible by 6, so a = 12 does not work.

-Jay
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by Jay@ManhattanReview » Thu May 18, 2017 11:36 am
rsarashi 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

OAA

It is given that both a and b are divisible by 6.

1) a = 2b + 6

If I put b=6, then a will be 18, which means 6 is the GCF -- YES

If I put a=6, then b will not be divisible by 6, so that means 6 will not be the GCF -- NO.


So how can A be the answer.

Please explain.

Thanks.
Let's take an Algebraic route to this question.

We have a and b are positive integers divisible by 6. Say a = 6c and b = 6d, where c and d are integers.

If c and d are co-prime, the answer is yes, the greatest common divisor of a and b is 6. Co-primes do not share a common factor between them except 1.

Statement 1: a = 2b + 6

=> 6c = 12d + 6

=> c = 2d + 1

The numbers d and (2d+1) are co-prime. Thus, the answer is yes, the greatest common divisor of a and b is 6.

You may test few values. Since (2d+1) is an odd number, you must not check with even values of d.

d = EVEN and (2d+1) = ODD are always co-prime.

Say,

1. d = 1, (2d+1) = 3 => 1 and 3 are co-prime
2. d = 3, (2d+1) = 7 => 3 and 7 are co-prime
3. d = 5, (2d+1) = 11 => 5 and 11 are co-prime

Statement 2: a = 3b

Say

1. b = 6 => a = 18. GCD = 6. The answer is yes, the greatest common divisor of a and b is 6.

2. b = 12 => a = 36. GCD = 12. The answer is No, the greatest common divisor of a and b is NOT 6.

The correct answer: A

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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by [email protected] » Thu May 18, 2017 2:14 pm
rsarashi wrote: Thank you so much for your reply. I understood .

Just a quick question, we can not put a=12 because if we put, then b will be 3, and 3 is not divisible by 6. So this case is invalid right?

Please confirm if my understanding is correct.

Thanks.
Hi rsarashi,

That is correct! The prompt explicitly states that both A and B are POSITIVE INTEGERS that are divisible by SIX - so when TESTing VALUES, you have to make sure that you use values that fit those restrictions.

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