Is the integer 2b divisible by 6 ?

Hi,
asethi, you are not correct when you say that various values of b are there for which the equation is satisfied .hence not suff..
if all values of b satisfy the condition, it is suff..

rohit8865 and asethi,

the solution is .

(1) 8b is divisible by 3...
answer is yes if b is an integer.. but ut is not given anywhere that b is an integer..
if b is say 3/2.. 8b would be div by 3 but 2b will not be div by 6..
so statement 1 is insuff...

(2) 9b is divisible by 12 ..
clearly insuff..

combined
we know that b can not be (3 or some multiple of 3)/8 as then 9b will not be div by 12...
therefore b has to be an integer.. which is a multiple of 3 and 2 ... so 2b is div by 6..suff
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(1) 8b is divisible by 3.
b could be 3,6,9,12....Insuffcient

(2) 9b is divisible by 12
b could be 4,8,12, 16....Insuffcient

Combining both b is 12...Suffcient

Ans: C

I could not get..

for statement 1 if b (as per u) if b=3,6,9,12 then 2b is easily divisible by 3

8b is divisible by 3.
Which means that when b=3, 8*3 is divisible by 3, again when b=6 or b=9 or b=12 and so on. So b can have many values..

Hope this is clear.

I don't understand why it is C.

The question is IS 2b divisible by 6, this is a yes no DS, not a values, If we prove b is divisible by 3 then we can answer with yes or no.

1 - Clearly they are saying b is divisible by 3 so there is unique answer yes. -- AD
2 - Clearly b is divisible by 4, hence be could be 8 or 12, meaning yes or no. Hence A is the answer.

Hi,
it is not mentioned b is div by 3...
8b is div by 3....so b could be 3/8....
hence A is not suff....
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Is the integer 2b divisible by 6 ?

(1) 8b is divisible by 3.

(2) 9b is divisible by 12

there fore, 2b/6 => (2b)/(3*2) => b/3
So , need to know whether b is divisible by 3 or not...

using first statement
8b/3 = (2 *2 *2*b)/3 => b should be divisible by 3.
So statement 1 is sufficient .

But 2nd is not sufficient

Hi chetan2u i think answer is option A only. As in the question it is written 2B is an integer . and if B=3/8 then 2B will not be an integer. Correct me if im missing something

Hi sharma123,

you are absolutely correct in your observation that b cannot be 3/8..
but still answer will not be A..

reason.. 3/8 was taken just to show that b need not be integer..
take b=3/2... now 2b =3, which is an integer..
8b=12, which satisfies the statement 1 tht 8b is div by 3..
yet 2b, which is 3, is not div by 6
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from Ist statement: B is multiple of 3. So, 2b is multiple of 6. Hence 1st statement is sufficient.

I think answer should be A. Please give more detailed explanation.

How is this the solution, when in your previous post you just said that statement 1 is not sufficient?

You claimed that if b= 3/2, then statement "(1) 8b is divisible by 3" can still hold true, because 8b = 8(3/2) = 24/2 = 12, and 12 is divisible by 3. However, when you plug it back into the question stem, is "is 2b divisible by 6," then 2b = 2 (3/2) = 3, and 3 is not divisible by 6. However, if b = 3, then 8b = 8(3) = 24, which is divisible by 3, and when you plug is into the question stem, then 2b = 2(3) = 6, which is divisible by 6. Therefore, statement 1 is insufficient.

Statement (2), 9b is divisible by 12, means that b must multiple by 9 so that the products becomes a multiple of 12 (it may equal 12 as well). Note, that it must still result in 2b being an integer as stated in the question stem. So, if b=4, than 9b = 36, and 36 is divisible by 12, and 2b = 2(4) = 8, which is not divisible by 6. However, if b=12, then 9b=108, which is divisible by 12, and when we plug 12 for b in the question stem, 2b=2(12)=24, we find that 24 is divisible by 6. Therefore, statement 2 is insufficient.

When we combine statements, we find that b has to be a number that when it's multiplied by 8, it is divisible by 3, and when it's multiplied by 9, it is divisible by 12. Therefore, b must be a common multiple for 3 & 12.
b = 12, 24, 36, 48, etc..
If any of those value for b are multiplied by 2, then 2b is divisible by 6.

Therefore, both statement 1&2 are required for sufficiency.

Answer is C.

*please kudos"

Hi,
You are CORRECT that My initial post was the correct...
This was again asked and I must have been half asleep while solving it..
+1 Kudos..
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is the integer 2b divisible by 6 ?

(1) 8b is divisible by 3.

(2) 9b is divisible by 12


In the original condition, there is 1 variable(b), which should match with the number of equations. So you need 1 equation, which is likely to make D the answer.
For 1), b=3 -> yes, b=3/2 -> no, which is not sufficient.
For 2), b=4 -> no, b=12 -> yes, which is not sufficient.
When 1) & 2), in order to have 8b divided by 3 and 9b divided by 12, only b=integer is possible. Thus, C is the answer. (It doesn't mean that b is an integer as since 2b is an integer.)


 For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

It is given that 2b is an integer. so b is in the form of k/2 where k is any integer (2 *k/2=k an integer). so we need to prove k is divisible by 6.

condition 1. given 8b is divisible by 3 so 8*k/2=4k is divisible by 3 so k can be =3,6,9,12,15,18,24.... so not sufficient.

condition 2. 9b is divisible by 12.9* k/2= 9k/2 is divisible by 12 or 3k is divisble by 8...k can be 8,16,24 ....so not sufficient.

combining them k can be 24,48 and so on,.....so k is divisible by 6 as required..


if correct ,plz give kudos thnks

Hi,

it is flawed

if b= 3/8
2b = 6/8 = 3/4 and is not an integer

please correct !

Thanks, I had menioned in a later post that b=3/8 was just to give an example..

If b =3/2...
8b = 8*3/2 = 12 and 12 is div by 3...
2b = 2*3/2 = 3, 3 is an integer but not div by 6,.....
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Is the integer 2b divisible by 6 ?

(1) 8b is divisible by 3.

(2) 9b is divisible by 12

My way
(1) 8b is divisible by 3
I means that
8b=3q , q∈ℕ
b = 3/8q
But we need to have 2b ∈ℕ
so 2b = 2*3/8q= 3/4q, q∈ℕ
and if 2b is an integer
so q= 4k , k∈ℕ to have no fraction in 2b
finaly => 2b= 3/4q= 3/4* 4k = 3k
So we can have 2b a multiple of 3
But we do not know if it is a multiple of 6 !
INSUFF



(2) 9b is divisible by 12
Same way
9b=3r , r∈ℕ
b = 4/3r
But we need to have 2b ∈ℕ
so 2b = 2*4/3r= 8/3r, q∈ℕ
and if 2b is an integer
so r= 3k , k∈ℕ
finaly => 2b= 8/3r= 8/3* 3k = 8k
So we can have 2b a multiple of 8
But we do not know if it is a multiple of 6 !
INSUFF

(1+2)
2b have to be a multiple of 8 and of 3 so of 24
And 24 is divisible by 6 !
SUFF !

Let me take a job at this Question.
This is an among question testing our Knowledge of integer Properties.
Here is what i did in this one -->


Given data --> 2b is an integer.
There are two scenarios possible from 2b being an integer.
Case 1-> b is an integer.
Case 2-> b is of the form \(\frac{Integer}{2}\)

We are asked if 2b/6 is an integer or not i.e b is dividable by 3 or not.


Statement 1-->
8b/3 is an integer.
Let take two cases =>
Case 1-> b is an integer
In this case b must be a multiple of 3 as 8 is not a multiple of 3.
Hence b must be divisible by 3.
Case 2-> b is of the from Integer/2
Let b=5/2 => 8b/3 => 8*15/2/3 will be an integer.
But 15/2 is clearly not dividable by 3.

Hence not sufficient.


Statement 2-->
9b/12 is an integer
OR=> 3b/4 must be an integer.
So b cannot be of the form Integer/2 as for that case --> 3b/4 will not be an integer.
Hence b is an integer.
Also as 3 is not dividable by 4=> b must be a multiple of 4.
But it may or may not be a multiple of 3.
E.g=> 4 =>NO
12= YES.
Hence not sufficient.

Combing the two statements => As b is an integer => b must be divisible by 3.

Hence sufficient.

Hence C.
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Let me take a job at this Question.
This is an amazing question testing our Knowledge of integer Properties.
Here is what i did in this one -->


Given data --> 2b is an integer.
There are two scenarios possible from 2b being an integer.
Case 1-> b is an integer.
Case 2-> b is of the form \(\frac{Integer}{2}\)

We are asked if 2b/6 is an integer or not i.e b is dividable by 3 or not.


Statement 1-->
8b/3 is an integer.
Let take two cases =>
Case 1-> b is an integer
In this case b must be a multiple of 3 as 8 is not a multiple of 3.
Hence b must be divisible by 3.
Case 2-> b is of the from Integer/2
Let b=5/2 => 8b/3 => 8*15/2/3 will be an integer.
But 15/2 is clearly not dividable by 3.

Hence not sufficient.


Statement 2-->
9b/12 is an integer
OR=> 3b/4 must be an integer.
So b cannot be of the form Integer/2 as for that case --> 3b/4 will not be an integer.
Hence b is an integer.
Also as 3 is not dividable by 4=> b must be a multiple of 4.
But it may or may not be a multiple of 3.
E.g=> 4 =>NO
12= YES.
Hence not sufficient.

Combing the two statements => As b is an integer => b must be divisible by 3.

Hence sufficient.

Hence C.
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