**Vincen wrote:**A list contains twenty integers, not necessarily distinct. Does the list contain at least two consecutive integers?

(1) If any single value in the list is increased by 1, the number of different values in the list does not change.

(2) At least one value occurs more than once in the list.

C is the OA.

I got confused. Statement (1) says that all the numbers in the list are different. Is it correct? I need help. Please. Experts could you explain to me how can I solve this question?

Statement 1: If

any single value in the list is increased by 1, the number of different values in the list does not change.

Case 1: Say the list is {0, 2, 4, 6, ..., 38}. Assume that all are distinct.

Thus, the number of different values in the list = 20.

After increasing any single number, say '4' by 1, we get the list as {0, 2,

5, 6, ..., 38}.

Thus, the number of different values in the list = Same as before = 20.

The answer is No. The list does not contain at least two consecutive integers.

Case 2: Say the list is {

1, 2, 4, 6, ..., 38}. Assume that all are distinct and there is a set of consecutive integers.

Thus, the number of different values in the list = 20.

After increasing any single number, say '4' by 1, we get the list as {1, 2,

5, 6, ..., 38}.

Thus, the number of different values in the list = 20.

The answer is Yes. The list contains at least two consecutive integers (1 and 2).

No unique answer. Insufficient.

Statement 2: At least one value occurs more than once in the list.

Case 1: Say the set is {2,

2, 3, 6, ..., 38}. The answer is Yes. The list contains at least two consecutive integers (2 and 3).

Case 2: Say the set is {2, 2, 4, 6, ..., 38}. The answer is No. The list does not contain at least two consecutive integers.

Statement 1 & 2:

Case 1: Say the list is {

2, 2, 4, 6, ..., 38}. At least one value appears more than once (2).

Thus, the number of different values in the list = 19.

After increasing any single number, say '4' by 1, we get the list as {

2, 2,

5, 6, ..., 38}.

Thus, the number of different values in the list = Same as before = 19.

The answer is No. The list does not contain at least two consecutive integers.

Case 2: Say the list is {1,

2, 2, 4, 6, ..., 36}. At least one value appears more than once (2).

Thus, the number of different values in the list = 19.

After increasing any single number, say 2 by 1, we get the set as {1, 2,

3, 4, 6, ..., 38}.

Thus, the number of different values in the list = 19 + 1 =

20.

This is not a valid case; since after the increase, the number of different values in the list is NOT equal to that before the increase.

Thus, if we take at least one set of consecutive integers and a number that appears twice, it is not possible to abide by Statement 1. Thus, the list does not contain at least two consecutive integers.

The correct answer:

C
Hope this helps!

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