twenty integers

[scottxyoung]

this data suff. question is from CAT #6; I disagree with the answer, and think it should be E.

Posting the question, and my contention with the explanation/answer in red.
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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.

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

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

Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.

EACH statement ALONE is sufficient.

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

Explanation for 1+2 together being sufficient: (1) AND (2) SUFFICIENT: Statement 2 indicates that at least one value occurs twice; call that value a.

Statement 1 indicates that increasing any value in the list by 1 will not change the number of distinct values in the list. In this case, then, increasing one of the a values by one, to a + 1, will still leave you with a in the list (since there are at least two a values) as well as a + 1. The value of a + 1, then, must already have been in the original list; if it wasn’t, then you would have just added a new value without getting rid of an old value, and statement 1 forbids this.

For example, if the original list is {1, 1, 2, 4, 6}, then a = 1 and there are four distinct values in the list. Changing one of the 1’s to 2 makes the list {1, 2, 2, 4, 6} and there are still four distinct values. This list does contain two consecutive integers (1 and 2).

If the original list were {1, 1, 3, 5, 7}, then a = 1 and there are four distinct values in the list. Changing one of the 1’s to 2 makes the list {1, 2, 3, 5, 7}, but now there are 5 distinct values in the list! This is not allowed, according to statement 1.

As a result, whatever a is, a + 1 must also be in the original list. The original list must contain at least one pair of consecutive integers.

However, what if the set was (1,3,5,5,7)? This satisfies both conditions (1) and (2), but has no consecutive numbers. Whereas the set (1,1,1,2,2) satisfies both conditions as well, but DOES have consecutive numbers. Thus, even together the statements are still inconclusive, and the answer should (i think) be E. Can anyone explain why its C?

Thanks,
Scott

[NL]

However, what if the set was (1,3,5,5,7)? This satisfies both conditions (1) and (2), but has no consecutive numbers. Whereas the set (1,1,1,2,2) satisfies both conditions as well, but DOES have consecutive numbers. Thus, even together the statements are still inconclusive, and the answer should (i think) be E. Can anyone explain why its C?

Thanks,
Scott

Both of your sets don’t satisfy the statement (1):

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

(1,3,5,5,7)
--> the number of different values is 4
If 5 is increased by 1, the new list is (1,3,5,6,7)
--> the number of different values is 5. NOT satisfy (1)
(the list should be 1,3,5,5,6)

(1,1,1,2,2)
--> the number of different values is 2
If 2 is increased by 1, the new list is (1,1,1,2,3)
--> the number of different values is 3. NOT satisfy (1)
(the list should be 1,1,1,1,2)

(This question was super hard for me at the first time I encountered it)

[RonPurewal]

Scott, try changing one of your 5's.

[angelachan26]

but you can also satisfy both 1 and 2 by using non-consecutive integers, for example (1,3,5,5,7). This set contains at least one value occurring more than once, and if you add 1 to 3, you'd have (1,4,5,5,7) which still has the same number of distinct values.

So isn't the answer E?

[RonPurewal]

but you can also satisfy both 1 and 2 by using non-consecutive integers, for example (1,3,5,5,7). This set contains at least one value occurring more than once, and if you add 1 to 3, you'd have (1,4,5,5,7) which still has the same number of distinct values.

So isn't the answer E?

Right, but, if you step up one of the 5's instead, you get 1, 3, 5, 6, 7. Previously four different numbers, now five.

So, that group of numbers doesn’t satisfy the statement. Can’t use it to prove anything, since it doesn’t count.

[RonPurewal]

What's happening here is that you're misinterpreting the word "any".

Just think of how "any" works in everyday English on the street"”"”it's totally the same here.
E.g., if a woman says "I'd marry any guy in this room", that means that she'd say yes to ALL of them. If any guy asked her, she'd say yes.

Same thing here. The statement has to work with EVERY number in the room, not just one of them.

[CatherineN556]

hi Ron
I think A is sufficient to prove that the list does not contain any consecutive integer, since the first statement proves that
any two integers in the list(let's assume that they are a and b ), |a-b| does not equal to 1.
Could u pls explain this for me?
thank u so much

[RonPurewal]

the answer key presents, via specific examples, an explicit proof that statement 1 is insufficient.

have you read through the answer key?
if so, please let us know, specifically, what part of that explanation you don't understand.
thanks.