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# m05 #22

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Re: m05 #22 [#permalink]

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02 Aug 2012, 06:10
thevenus wrote:
Thanks a lot, Kudos for you +1
Wasn't the previous one was tougher? why did you changed / modified ? Now the GMAT can't put such an option (of choosing irrational no. at least if not negative numbers?)

Again, every GMAT divisibility question will tell you in advance that any unknowns represent positive integers, so the stem should have mentioned that x is a positive integer.
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Re: m05 #22 [#permalink]

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02 Aug 2012, 06:52
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Is X divisible by 15?
1. When X is divided by 10, the result is an integer
2. X is a multiple of 30

An important aspect of any data sufficiency question is to apply to the FOIN checks to ensure we coverall the different types of values that X may assume.
Many a time, when answering data sufficiency questions, we overlook the fact that the X may be ZERO, negative, fraction or an irrational number.

In order to establish that X is divisible by 15, we have to establish BOTH :-
a) X is an integer (Note : 0 is considered divisible by any integer)
b) X is divisible by BOTH 5 and 3.

Here are the FOIN checks :-
---- F - Can the number be a Fraction ?
---- 0 - Can the number be zero ?
---- I - Can the number be an Integer or an irrational number ?
---- N - Can the number be negative ?

Lets consider the answer options and apply the FOIN
a) When n is divided by 10, the result is an integer

F - Can X be a fraction ? No. Any fraction divided by 10 leads to yet another fraction
0 - Can X be zero. Yes it can.. ZERO Divided by anything is an integer.
I - Can X be an integer or an irrational number.... No...X has to be an integer , as any irrational number divided by a
integer can only lead to another irrational number
N - Can X be negative - Yes it can. But this fact doesnt bother us too much as we dont care about whether X is negative
or positive, just the fact whether X is a multiple of 15.

So from 1) we have established that X is an integer and that X is divisible by 10.
But wait, if X is divisible by 10, X is divisible by all PRIME factors of 10. This implies that X is divisible by
2,5 (all PRIME factors of 10). We have not been able to conclusively establish that X is also divisible by 3.
For e.g : X may be 100 or 300. Both 100 and 300 are divisible by 10, but only 300 is divisible by 3.
HENCE 1) on its own is INSUFFICIENT...

Lets take 2
X^2 is a multiple of 30
When we apply the FOIN test, we find out that X may be an irrational number.
X can be SQRT(30).
SQRT(30) * SQRT(30) = 30, which is divisible by 30.

Hence lets not bother with further checks as this statement is clearly insufficient on its own. Remember we need to
prove that X is an integer.

Let's for a second evaluate the statement " X ^ 2" is multiple of 30. Let's understand the implication of this statement.
For this discussion, let's momentarily assume that X is an integer.
" X ^ 2" is multiple of 30 ======> X^2 has 30 as its factor==>X^2 has all factors of 30 as its factor
Therefore X^2 has 2,3,5 as its PRIME factors.
Think about this for a minute, we get X^2 when we multiple X * X.
Hence if we look at the PRIME factors of X^2, we should see each PRIME factor of X occurring 2 times.
Look at an example
Lets say X = 6 . The PRIME factors of X are 2,3.
X * X = 36
Factors of 36 should include two occurrences of 2 and two occurrences of 3.
which makes sense as 36 = 2 *2 * 3*3.

So what this implies is that if (X ^2) has 2,3,5 as its factors, then X should also have 2,3,5 as its PRIME factors
If we knew for sure that X was an integer, this statement would have been sufficient on its own. However, we dont know for
sure that X is an integer.
Hence (2) on its own is not sufficient.

1 + 2 ==> X is an integer and that X has 2,3,5 as its factors.
Hence (C)
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Re: m05 #22 [#permalink]

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02 Aug 2012, 10:25
Bunuel wrote:
Next, every GMAT divisibility question will tell you in advance that any unknowns represent positive integers (ALL GMAT divisibility questions are limited to positive integers only). So, we edited this question and in the new GMAT Club tests this question reads:

If x is a positive integer, is x divisible by 15?

(1) x is a multiple of 10 --> if $$x=10$$, then the answer is NO but if $$x=30$$, then the answer is YES. Not sufficient

(2) x^2 is a multiple of 12 --> since $$x$$ is an integer, then $$x^2$$ is a perfect square. The least perfect square which is a multiple of 12 is 36. Hence, the least value of $$x$$ is 6 and in this case the answer is NO, but if for example $$x=12*15$$ then the answer is YES. Not sufficient.

Notice that from this statement we can deduce that $$x$$ must be a multiple of 3 (else how can this prime appear in $$x^2$$?).

(1)+(2) $$x$$ is a multiple of both 10 and 3, hence it's a multiple of 30, so $$x$$ must be divisible by 15. Sufficient.

Answer: C.

Hope it's clear.

Just thought I'd take a crack at the expanding on option B. I was left with my head scratching for a while when I read your explanation of choice B, but I worked it out (I hope so!)

$$x^2$$ is a multiple of 12.

Now $$12 = 3*2*2$$

Therefore $$x^2 = 3*2*2*m$$

$$m$$ must at least be a 3. Why? Because the prime numbers need to repeat at least once since we are multiplying the number by itself when we square it.

e.g.
$$6=3*2$$
$$6^2=3*2*3*2$$

Therefore, in order for x to multiply by itself, we need $$m$$ to equal at least a 3. In which case $$x^2=36$$ and x=6. In which case $$\frac{6}{15}$$ does not work!

However, if $$m=15$$, then the statement is sufficient. That's why B by itself is insufficient.

The correct answer of the modified question is therefore C.
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Re: m05 #22 [#permalink]

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02 Aug 2012, 16:47
I went with (B) and realized my mistake. I should have plugged in more numbers so as to eliminate B.
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Re: m05 #22 [#permalink]

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02 Aug 2013, 04:37
Well, its C.

I always end up messing with the premise is X an integer.

If X is an integer, the answer is B.
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Re: m05 #22 [#permalink]

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03 Aug 2013, 03:45
it went for me in this way :

For x to be divisible my 15 - should have atleast one 3 and atleast one 5.

Statement 1 - stated x/10 is an integer, which means x is properly divisible by ten, whose factors are 2 and 5. Not sufficient alone. - gave me a 5

Statement 2 - stated x/30 is an integer hence by same logic factor of 30 are 2*3*5. But this has a twist that as 15 is smaller than 30 so 30 cannot divide 15 with 0 remainder.

Hence (1)+(2) - guides me to following conclusions :
1. We have ample number of factors to qualify.
2. It is smaller 30 and greater than 10.

Correct me, if I am wrong.
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Re: m05 #22 [#permalink]

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11 Aug 2013, 01:10
BELOW IS REVISED VERSION OF THIS QUESTION:

If x is a positive integer, is x divisible by 15?

(1) x is a multiple of 10 --> if x=10 then the answer is NO but if x=30 then the answer is YES. Not sufficient

(2) x^2 is a multiple of 12 --> the least perfect square which is a multiple of 12 is 36. Hence, the least value of $$x$$ is 6 and in this case the answer is NO, but if for example $$x=12*15$$ then the answer is YES. Not sufficient.

Notice that from this statement we can deduce that $$x$$ must be a multiple of 3 (else how can this prime appear in $$x^2$$?).

(1)+(2) $$x$$ is a multiple of both 10 and 3, hence it's a multiple of 30, so $$x$$ must be divisible by 15. Sufficient.

Answer: C.
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Re: m05 #22 [#permalink]

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18 Jul 2014, 07:56
Question: Is X divisible by 15?

S1: When X is divided by 10, the result is integer.
X = 20 => No to the question.
X = 30 => Yes to the question.
So, S1 is not sufficient. Eliminate A and D.

S2: X^2 is a multiple of 30.
X^2 = 60 => X = Square root of 60 => No to the question.
X^2 = 900 => X = 30 => Yes to the question.
So, S2 is not sufficient. Eliminate B.

Both S1 and S2:
If X is divisible by 10 then X^2 must at least be divisible by 100.
Additionally X^2 is multiple of 30.

Thus applying both S1 and S2, X^2 must be divisible by 300.
X^2 = 300 => X = Square root of 300 => No to the question.
X^2 = 900 => X = 30 => Yes to the question.
Therefore, Both S1 and S2 are also not sufficient.

Eliminate C. Correct answer is E.
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Re: m05 #22 [#permalink]

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18 Jul 2014, 08:16
Let’s look at a different scenario: Let’s say the question includes a constraint that X is an integer and suppose that the two given statements remain unchanged.

Modified Question: Given that X is an integer, is X divisible by 15?

S1: When X is divided by 10, the result is integer.
X = 20 => No to the modified question.
X = 30 => Yes to the modified question.
So, S1 is not sufficient. Eliminate A and D.

S2: X^2 is a multiple of 30.
Per modified question if X is an integer, X^2 must be a perfect square that is a multiple of 30.
In such case the data such as X^2 = 30, 60, 90, 120, 180, and others which are not perfect square become invalid.

So, we are left with data such as X^2 = 900, 3600, 8100 which are the perfect square and are multiple of 30.
X^2 = 900 => X = 30 => Yes to the modified question.
X^2 = 3600 => X = 60 => Yes to the modified question.
X^2 = 8100 => X = 90 => Yes to the modified question.

So, S2 is sufficient. Correct answer is B for the above stated modified question.
Re: m05 #22   [#permalink] 18 Jul 2014, 08:16

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# m05 #22

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