Number Properties

[Anahatha]

If 9 is a factor of 2x , is there is a chance for (6x/54)+(2x/3) to be NOT AN INTEGER ? I answered the that it will always be an integer. Is it right? Please explain....

[manpsingh87]

If 9 is a factor of 2x , is there is a chance for (6x/54)+(2x/3) to be NOT AN INTEGER ? I answered the that it will always be an integer. Is it right? Please explain....

well yes.. your answer is correct..!!
as 9 is a factor of 2x that means x will be a factor of 9..!!!
now consider 6x/54 it can re written as x/9; as 9 is a factor of x therefore x/9 will always be an integer..!!

2x/3; as 9 is a factor of 9 therefor x/3 will always be an intger;

we also know that when an integer is added to another integer it will result in integer as well...!!

hence their sum will be an integer as well...!!!

[Anahatha]

Thanks for the explanation. I too reasoned the same way. But please have a look at this question from Kaplan Test:




Is the answer given by Kaplan wrong or am I missing something??

[Stuart@KaplanGMAT]

Thanks for the explanation. I too reasoned the same way. But please have a look at this question from Kaplan Test:




Is the answer given by Kaplan wrong or am I missing something??

Hi!

The question is correct.

Both of you assumed that x has to be an integer, something not stipulated by the question.

Since x doesn't have to be an integer, one cannot assume that if 9 is a factor of 2x, 9 must also be a factor of x.

For example, if you plug in x=4.5, you get:

9 is a factor of 2x, i.e. 9 is a factor of 9; and

6x/54 + 2x/3 = 27/54 + 9/3 = 1/2 + 3 = 3.5 which is NOT an integer.

In fact, the simplest way to solve this type of question is to plug in numbers; pick the smallest possible value for x (in this case 4.5) and plug it into the choices until you find one that isn't an integer.

[abhi0697]

IMO, the answer from Kaplan is not wrong as:
6x/54 + 2x/3 = 3/6 * 2x/9 + 2x/3

Now when 9 is a factor of 2x; it means 9 is a factor of x when x is an integer; and 9 has to be factor of 2x when x is not an integer (for example, x = 22.5)

So, 3/6 will be cancelled by 2 (which comes from 2x divided by 9) only when x is integer. When x is not an integer then 2x has to be integer to have 9 as a factor.

Hence, 6x/54 + 2x/3 will not always be integer given that 9 is a factor of 2x.

[manpsingh87]

Thanks for the explanation. I too reasoned the same way. But please have a look at this question from Kaplan Test:




Is the answer given by Kaplan wrong or am I missing something??

Hi!

The question is correct.

Both of you assumed that x has to be an integer, something not stipulated by the question.

Since x doesn't have to be an integer, one cannot assume that if 9 is a factor of 2x, 9 must also be a factor of x.

For example, if you plug in x=4.5, you get:

9 is a factor of 2x, i.e. 9 is a factor of 9; and

6x/54 + 2x/3 = 27/54 + 9/3 = 1/2 + 3 = 3.5 which is NOT an integer.

In fact, the simplest way to solve this type of question is to plug in numbers; pick the smallest possible value for x (in this case 4.5) and plug it into the choices until you find one that isn't an integer.

yep.. agreed... sir.. thanks for the explanation...,, i will be more careful in future while tackling such problems...!!!!

[Anahatha]

Thanks a lot for the explanations . Really helpful .