What is the remainder when the positive integer x is divided

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)