Number Properties

Anahatha wrote: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...!!!

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??

Anahatha wrote: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.

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.

Stuart Kovinsky wrote:

Anahatha wrote: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...!!!!

Thanks a lot for the explanations . Really helpful .