Integers Consecutive - DS

[karthikpandian19]

The mean of a set of n consecutive positive integers is k. One of the integers is removed from the set to create a new set of positive integers with a mean of m. Is n even?

(1) k = m

(2) The least element of the original set of integers is odd.

[shankar.ashwin]

A IMO

If k=m, the only possibility I can think of it the set containing odd number of integers and the number removed is the mean itself.

For eg: (1,2,3) mean = 2. Now if 2 is removed the set becomes (1,3) with mean = 2

Similarly consider (1,2,3,4,5) mean = 3. With 3 removed the set becomes (1,2,4,5) with mean = 3

(or) (4,5,6) etc

Therefore, for such a condition the set should contain 'odd' number of consecutive integers. - 'n' cannot be even. (Sufficient)

(2) doesn't give much info,

[GmatMathPro]

A IMO

If k=m, the only possibility I can think of it the set starting with 1 (or any add number) and ending in a odd number and the number removed is the mean itself.

For eg: (1,2,3) mean = 2. Now if 2 is removed the set becomes (1,3) with mean = 2

Similarly consider (1,2,3,4,5) mean = 3. With 3 removed the set becomes (1,2,4,5) with mean = 3

(or) (3,4,5) etc

Therefore, for such a condition the set should contain 'odd' number of consecutive integers.

(2) doesn't give much info,

The starting and ending numbers wouldn't necessarily have to be odd though. For example (4,5,6) mean =5, (4,6) mean=5. The key ,as you mention, is that removing a number from a set of numbers will always change the mean unless that number is exactly equal to the mean itself. But the mean will only be an element of the set if we have an odd number of consecutive integers, so that is only a possibility if n is odd.

[apex231]

But the mean will only be an element of the set if we have an odd number of consecutive integers, so that is only a possibility if n is odd.

Hi GmatMathPro, can you explain this?

Thanks!

[GmatMathPro]

But the mean will only be an element of the set if we have an odd number of consecutive integers, so that is only a possibility if n is odd.

Hi GmatMathPro, can you explain this?

Thanks!

The mean of a set of consecutive integers is always equal to the median. If we have an odd number of consecutive integers, the median and mean will both be the integer in the middle. For example, 4,5,6,7,8: The median and mean is 6. However, if we have an even number of consecutive integers, such as 5,6,7,8, then the median and mean is the average of the two middle numbers, 6.5, but 6.5 is not in the set of numbers. To have a middle number, we have to have an odd number of consecutive integers. If we have an even number of consecutive integers, the mean will always be the average of the two middle numbers, which will never be an integer, and hence cannot be an element of the set of consecutive integers.

Does that help?

[apex231]

Yes, it helps. Thanks for explaining.

But the mean will only be an element of the set if we have an odd number of consecutive integers, so that is only a possibility if n is odd.

Hi GmatMathPro, can you explain this?

Thanks!

The mean of a set of consecutive integers is always equal to the median. If we have an odd number of consecutive integers, the median and mean will both be the integer in the middle. For example, 4,5,6,7,8: The median and mean is 6. However, if we have an even number of consecutive integers, such as 5,6,7,8, then the median and mean is the average of the two middle numbers, 6.5, but 6.5 is not in the set of numbers. To have a middle number, we have to have an odd number of consecutive integers. If we have an even number of consecutive integers, the mean will always be the average of the two middle numbers, which will never be an integer, and hence cannot be an element of the set of consecutive integers.

Does that help?

[karthikpandian19]

Ashwin,

I had tried this and it is not required that the integers set need to end with ODD number. Either way it works

A IMO

If k=m, the only possibility I can think of it the set starting with 1 (or any add number) and ending in a odd number and the number removed is the mean itself.

For eg: (1,2,3) mean = 2. Now if 2 is removed the set becomes (1,3) with mean = 2

Similarly consider (1,2,3,4,5) mean = 3. With 3 removed the set becomes (1,2,4,5) with mean = 3

(or) (3,4,5) etc

Therefore, for such a condition the set should contain 'odd' number of consecutive integers.

(2) doesn't give much info,

[shankar.ashwin]

Ashwin,

I had tried this and it is not required that the integers set need to end with ODD number. Either way it works

Yeah Pete mentioned that previously aswell.. I clearly didn't check for all possibilities. The answer remains the same, I'll probably edit my post.