albany09 wrote:

What is the remainder when the positive integer x is divided by 8?

(1) When x is divided by 12, the remainder is 5.

(2) When x is divided by 18, the remainder is 11.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.

There is a very clean approach to solve such questions within a minute and without writing down anything. But for that you need to understand 'Divisibility' well. If you are willing to do that, read these 4 posts first:

http://www.veritasprep.com/blog/2011/04 ... unraveled/http://www.veritasprep.com/blog/2011/04 ... y-applied/http://www.veritasprep.com/blog/2011/05 ... emainders/http://www.veritasprep.com/blog/2011/05 ... s-part-ii/Now let's look at the question:

What is the remainder when the positive integer x is divided by 8?

This means: What is leftover when you make groups of 8?

Statement 1: When x is divided by 12, the remainder is 5.

When you make groups of 12, 5 balls are leftover. When you make groups of 8 instead, each of the groups of 12 balls leaves 4 balls. If no. of groups of 12 is even, you can combine 2 groups of 4 balls each to make more groups of 8. In that case, 5 balls will be still leftover. So a remainder of 5 is possible.

If no. of groups of 12 is odd, 4 balls will be leftover from one group of 12 and 5 balls will be still leftover. So a total of 9 balls will be leftover. We can make another group of 8 out of these 9 balls and 1 ball will be leftover. So a remainder of 1 is also possible.

Since remainder can be 5 or 1, this statement alone is not sufficient.

Statement 2: When x is divided by 18, the remainder is 11.

When you make groups of 18, 11 balls are leftover. When you make groups of 8 instead, each of the groups of 18 balls makes 2 groups of 8 balls each and leaves 2 balls.

Now there are 4 possibilities:

1. We are left with 2 balls + the original 11 remaining balls = 13 balls

When you make another group of 8 from 13, remainder will be 5

2. We are left with 2+2 balls + the original 11 remaining balls = 15 balls

When you make another group of 8 from 15, remainder will be 7

3. We are left with 2+2+2 balls + the original 11 remaining balls = 17 balls

When you make 2 groups of 8 from 17, remainder will be 1

4. We are left with no groups of 2 balls since they all make a complete group of 8. Only the original 11 balls are remaining. When you make a group of 8 from 11, remainder will be 3.

Since remainder can be 5, 7, 1 or 3, this statement alone is not sufficient.

Using both statements, remainder can be either 5 or 1 so they both together are not sufficient.

Answer (E)

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