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

Question

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

(1)  x  is a multiple of 10
 
(2)  x^2  is a multiple of 12

M05-22

Is x divisible by 15?

(1) When x is divided by 10, the result is an integer
(2) x^2 is a multiple of 30

gmat1220 · Answer

1) x=10 the answer is No
X=30 the answer is yes

2) x^2=900 the answer is yes
X^2=30 the answer is no

1) +2) means x^2 is multiple of 30 where x is an integer. 30 has primes 2,3,5. Hence x is multiple of 15. Hence C

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victorxman182 · Answer

Is X divisible by 15?   Does X contain 3 and 5 as prime factors?

1. When X is divided by 10, the result is an integer
   X contains 2 and 5 as prime factors, no info about 3. Insufficient.  
2. X^2 is a multiple of 30
    30=(3)(5)(2) which means that X at least contains these factors since we are dealing with unique factors. Sufficient.

https://i.imgur.com/GgSe5.png

Answer B.

What am I doing wrong?

gmat1220 · Answer

I think you are over assuming that x is integer in 2) when it is not given. So you will have to combine both the statements

Posted from my mobile device

fluke · Answer

Also discussed here:
 https://gmatclub.com/forum/m05-70531.html 
 https://gmatclub.com/forum/is-x-divisible-by-75300.html

pesfunk · Answer

Nice catch !!

Even I was trapped by assuming that it was an integer...never though that it could have been a root(30) as well :'(

IanStewart · Answer

Unless x is an integer, the question "is x divisible by 15?" is one you can only answer if you know a subject in advanced mathematics known as 'ring theory' (a subject which extends the concept of divisibility beyond integers). That's millions and millions of miles beyond the scope of the GMAT. Technically the answer should be B to this question if you know advanced math, but as a GMAT question, this problem simply doesn't make any sense. GMAT divisibility questions are  always  restricted to integers only; you will  never  be asked "is x divisible by 15" on the GMAT without being told, in advance, that x is an integer. The 'trap' in this question (that x might not be an integer) is simply ridiculous, and is definitely not the kind of trap you will ever find on the actual test. It's a very poorly designed practice question (though it would be fine if they told you x was an integer in the question stem, in which case the answer is B), and if you answered B here, then you have the right answer as far as the GMAT is concerned. Move on to more realistic practice material.

PallavTandon · Answer

You cannot assume x is an integer. This is one of the common catches in the GMAT questions

IanStewart · Answer

It can be a 'trap' in certain abstract algebra questions, but I can guarantee you that it is absolutely never a 'trap' in a divisibility question. In every single GMAT divisibility question you will ever see with unknowns in it, the question will tell you that those unknowns represent integers. The question "Is x divisible by 15" makes no sense if x is not an integer, because for GMAT purposes, the concept of divisibility is only defined for integers. GMAT questions always make sense, so they need to tell you x is an integer.

PallavTandon · Answer

Thanks Ian. If it does not mention integer, I am sure the question is incomplete. However would be it advisable to assume an integer in these questions.

ksp · Answer

Must have mentioned X is an integer.... Drained my 10 minutes. After which got frustrated and stopped the test and took a break. Lol.. :p

santorasantu · Answer

I have a following doubt, everybody is going for C for the answer and I think it is E.

the approach is as follows:

from 1: when x is divided by 10, the result is an integer. e.g. x= 20, 30, 40.....any number that ends in 0. In case of 30, x is divisible by 30 but when x = 20, then not,2 answers so NSF
from 2: x^2 is a multiple of 30. there for x^2 can be 60 or 90 or 120 or 150....which implies that x is sqrt(60) or sqrt(90) or sqrt (120)...when x is sqrt(60), then not divisible by 15, but when x = sqrt(900) = 30 ,then divisible by. therefore NSF.

combining 1 + 2: suppose x^2 is 60000 which implies x is sqrt(60000) = 100*sqrt(6). This number is divisible by 10 as per statement 1 and x^2 is a multiple of 30 as per statement 2. but not divisible by 15.
suppose x^2 is 900, then we have an answer that x is divisible by 15.
so 1+2 is also NSF.

I go for E, please let me know if I missed something

Bunuel · Answer

\sqrt{60,000}=100\sqrt{6}  is NOT divisible by 10.

1. 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).

2. On the GMAT when we are told that  a  is divisible by  b  (or which is the same: " a  is multiple of  b ", or " b  is a factor of  a "), we can say that:
(i)  a  is an integer;
(ii)  b  is an integer;
(iii)  \frac{a}{b}=integer .

So,  \sqrt{60,000}=100\sqrt{6}  is NOT divisible by 10, because  \sqrt{60,000}=100\sqrt{6}  is NOT an integer and the result is also NOT an integer.

Is  x  divisible by 15?

(1) When  x  is divided by 10, the result is an integer -->  \frac{x}{10}=integer  -->  x=10*integer . Now, if  x=0  (in case  integer=0 ), then the answer is YES but if  x=10  (in case  integer=1 ), then the answer is NO. Not sufficient.

From this statement though we can deduce that  x  is an integer (since  x=10*integer=integer ).

(2)  x^2  is a multiple of 30 --> if  x=0 , then the answer is YES but if  x=\sqrt{30} , then the answer is NO. Not sufficient.

(1)+(2) Since from (1)  x=integer  then in order  x^2  to be divisible by 30=2*3*5,  x  must be divisible by 30 ( x  must be a multiple of 2, 3 and 5, else how can this primes appear in  x^2 ?), hence  x  is divisible by 15 too. Sufficient.

Notice that  x  can be positive, negative or even zero, but in any case it'll be divisible by 30.

Answer: C.

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. 
 
 x  could be 10 (not divisible by 15) or 30 (divisible by 15). Thus, this statement alone is insufficient.
 
(2)  x^2  is a multiple of 12. 
 
Since  x  is a positive integer,  x^2  is a perfect square. The smallest perfect square that is a multiple of 12 is 36, which implies that the least value of  x  is 6. However, if  x=6 , then it is not divisible by 15, while if  x=6*15 , it is divisible by 15. Therefore, this statement alone is insufficient.
 
Note that from statement (2), we can deduce that  x  must be a multiple of 3, as otherwise, the prime factor of 3 would not appear in  x^2 .
 
(1)+(2) Since  x  is a multiple of both 10 and 3, it must be a multiple of their least common multiple, which is 30. Thus,  x  is divisible by 15. Therefore, the combination of both statements is sufficient to answer the question.

Answer: C

Hope it's clear.

santorasantu · Answer

Thanks a lot for the detailed explanation.

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

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